HOL html


(* ========================================================================== *)
(* FLYSPECK - BOOK FORMALIZATION                                              *)
(*                                                                            *)
(* Chapter: Fan                                              *)
(* Author: Hoang Le Truong                                        *)
(* Date: 2010-02-09                                                           *)
(* ========================================================================== *)



flyspeck_needs "general/sphere.hl";;
flyspeck_needs "fan/introduction.hl";;
flyspeck_needs "fan/topology.hl";;
flyspeck_needs "fan/fan_misc.hl";;


module Planarity  = struct


open Sphere;;
open Fan_defs;;
open Hypermap;;
open Vol1;;
open Fan;;
open Topology;;		
open Fan_misc;;

(***************************************************************************)
(**********************DHVFGBC                      ************************)
(***************************************************************************)



let collinear_continuous_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 c:real.
(\(t:real^1). (&1- drop(t))%u + drop(t) %w - (&1 -c)%x - c% v) continuous_on (:real^1)`,
  SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT;OPEN_UNIV;DIMINDEX_1]
THEN REPEAT STRIP_TAC
THEN REWRITE_TAC[drop]
THEN MATCH_MP_TAC CONTINUOUS_ADD 
THEN STRIP_TAC
THENL[
 MATCH_MP_TAC CONTINUOUS_VMUL 
THEN REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS1]
THEN MATCH_MP_TAC REAL_CONTINUOUS_SUB
THEN SIMP_TAC[REAL_CONTINUOUS_CONST;REAL_CONTINUOUS_AT_COMPONENT; DIMINDEX_1; ARITH];
REPEAT(MATCH_MP_TAC CONTINUOUS_SUB 
THEN SIMP_TAC[CONTINUOUS_CONST])
THEN MATCH_MP_TAC CONTINUOUS_VMUL 
THEN REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS1]
THEN SIMP_TAC[REAL_CONTINUOUS_AT_COMPONENT; DIMINDEX_1; ARITH]]);;

let collinear1_continuous_fan=prove(`!u:real^3 w:real^3 t:real^1.
(\(t:real^1). (&1- drop(t))%u + drop(t) %w) continuous at t`,
REPEAT STRIP_TAC
THEN REWRITE_TAC[drop]
THEN MATCH_MP_TAC CONTINUOUS_ADD 
THEN STRIP_TAC
THENL[
 MATCH_MP_TAC CONTINUOUS_VMUL 
THEN REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS1]
THEN MATCH_MP_TAC REAL_CONTINUOUS_SUB
THEN SIMP_TAC[REAL_CONTINUOUS_CONST;REAL_CONTINUOUS_AT_COMPONENT; DIMINDEX_1; ARITH];
 MATCH_MP_TAC CONTINUOUS_VMUL 
THEN REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS1]
THEN SIMP_TAC[REAL_CONTINUOUS_CONST;REAL_CONTINUOUS_AT_COMPONENT; DIMINDEX_1; ARITH]]);;


let CONTINUOUS_CLOSED_PREIMAGE_CONSTANT = prove
 (`!f:real^M->real^N s a.
      f continuous_on s /\ closed s ==> closed {x | x IN s /\ f(x) = a}`,
  REPEAT STRIP_TAC THEN
  ASM_CASES_TAC `{x | x IN s /\ (f:real^M->real^N)(x) = a} = {}` THEN
  ASM_REWRITE_TAC[CLOSED_EMPTY] THEN ONCE_REWRITE_TAC[SET_RULE
   `{x | x IN s /\ f(x) = a} = {x | x IN s /\ f(x) IN {a}}`] THEN
  MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE THEN
  ASM_REWRITE_TAC[CLOSED_SING] THEN  SET_TAC[]);;
let open_collinear_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 c:real.
open{t| ~((\(t:real^1). (&1- drop(t))%u + drop(t) %w - (&1 -c)%x - c% v)(t)= vec 0)}`,
REPEAT STRIP_TAC
THEN REWRITE_TAC[OPEN_CLOSED;DIFF; IN_ELIM_THM;]
THEN MP_TAC(ISPECL[`(\(t:real^1). (&1- drop(t))%u + drop(t) %w - (&1 -c)%x - c% v:real^3)`;`(:real^1)`;
`((vec 0):real^3)`]CONTINUOUS_CLOSED_PREIMAGE_CONSTANT)
THEN SIMP_TAC[CLOSED_UNIV; DIMINDEX_1; collinear_continuous_fan]);;

let open_vector_angle_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 c:real a:real.
(!t. ~((&1 - t) % u + t % w = x))
==>
open{t|  ~((\(t:real^1). vector_angle (v - x) (((&1 - drop(t)) % u + drop(t) % w) - x))(t) = a)}`,
REPEAT STRIP_TAC
THEN REWRITE_TAC[OPEN_CLOSED;DIFF; IN_ELIM_THM;]
THEN MP_TAC(ISPECL[`lift o (\(t:real^1). vector_angle (v - x:real^3) (((&1 - drop(t)) % u + drop(t) % w) - x))`;`(:real^1)`;
`lift (a:real)`]CONTINUOUS_CLOSED_PREIMAGE_CONSTANT)
THEN REWRITE_TAC[o_DEF;LIFT_EQ]
THEN DISCH_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN SIMP_TAC[CLOSED_UNIV; DIMINDEX_1;]
THEN REWRITE_TAC[GSYM o_DEF]
THEN REWRITE_TAC[GSYM FORALL_LIFT_FUN]
THEN MP_TAC(ISPECL[`x:real^3 `;`v:real^3 `;`u:real^3`;` w:real^3`;` &0`]collinear_continuous_fan)
THEN REDUCE_ARITH_TAC
THEN REDUCE_VECTOR_TAC
THEN SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT;OPEN_UNIV;DIMINDEX_1]
THEN MATCH_MP_TAC MONO_FORALL
THEN GEN_TAC
THEN DISCH_THEN (LABEL_TAC"MA")
THEN STRIP_TAC
THEN REMOVE_THEN "MA" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN DISCH_TAC
THEN MP_TAC(ISPECL[`(\(t:real^1). ((&1- drop(t))%(u:real^3) + drop(t) %(w:real^3)) - (x:real^3) )`;`(\(t:real^3). lift (vector_angle ((v:real^3)-(x:real^3)) t))`;`x':real^1`] CONTINUOUS_AT_COMPOSE)
THEN ASM_REWRITE_TAC[GSYM o_ASSOC]
THEN REWRITE_TAC[o_DEF]
THEN DISCH_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(A+B)-C=A+B-C:real^3`;GSYM(o_DEF)]
THEN REWRITE_TAC[GSYM(REAL_CONTINUOUS_CONTINUOUS1);GSYM(I_DEF);I_O_ID]
THEN MATCH_MP_TAC(ISPECL[`(v:real^3)-(x:real^3)`;`(&1 - drop x') % u + drop x' % w - x:real^3
`]REAL_CONTINUOUS_AT_VECTOR_ANGLE)
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A+B-C= vec 0<=> A+B=C:real^3`]);;



let exists_open_not_collinear=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E
==> 
?t1:real. &0 < t1 /\ t1 <= &1
/\ (!t:real. &0<= t /\ t<= t1==> ~(collinear{x,v,(&1-t)%u+ t % w}))`,

REPEAT STRIP_TAC
THEN REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN ONCE_REWRITE_TAC[COLLINEAR_3;]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` u:real^3`;`v:real^3`]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;
`(E:(real^3->bool)->bool)`;` w:real^3`;` u:real^3`]
THEN FIND_ASSUM MP_TAC`~collinear {(x:real^3),(u:real^3),(w:real^3)}`
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`{A,B,C}={B,C,A}`;]
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[collinear_fan]
THEN ASM_REWRITE_TAC[aff; AFFINE_HULL_2;IN_ELIM_THM]
THEN DISCH_THEN(LABEL_TAC"MA")
THEN FIND_ASSUM MP_TAC`~((u:real^3) IN aff {(x:real^3), (v:real^3)})`
THEN REWRITE_TAC[aff; AFFINE_HULL_2;IN_ELIM_THM]
THEN DISCH_THEN(LABEL_TAC"A")
THEN SUBGOAL_THEN`!(t:real). ~((&1 - t) % u + t % w = x:real^3)`ASSUME_TAC
THENL(*1*)[
GEN_TAC
THEN REMOVE_THEN "MA" MP_TAC
THEN MATCH_MP_TAC MONO_NOT
THEN DISCH_TAC
THEN EXISTS_TAC`&1-(t:real)`
THEN EXISTS_TAC`(t:real)`
THEN ASM_REWRITE_TAC[REAL_ARITH`&1 - t + t = &1`];(*1*)


SUBGOAL_THEN`!(t:real). (collinear {vec 0, v - x, ((&1 - t) % u + t % w) - x} <=>
      vector_angle (v - x:real^3) (((&1 - t) % u + t % w) - x) = &0 \/
      vector_angle (v - x) (((&1 - t) % u + t % w) - x) = pi)`ASSUME_TAC
THENL(*2*)[

GEN_TAC
THEN MP_TAC(ISPECL[`(v:real^3)-(x:real^3)`;`((&1 - t) % (u:real^3) + (t:real) % (w:real^3)) - (x:real^3)`]
COLLINEAR_VECTOR_ANGLE)
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A-B=vec 0<=> A=B`];(*2*)

ASM_REWRITE_TAC[]
THEN MRESA_TAC open_vector_angle_fan[ `x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`;`c:real`;`&0`]
THEN MRESA_TAC open_vector_angle_fan[ `x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`;`c:real`;`pi`]
THEN REWRITE_TAC[DE_MORGAN_THM;]
THEN MP_TAC(ISPECL[`{(t:real^1) | ~(   
vector_angle ((v:real^3) - x) (((&1 - drop(t)) % (u:real^3) + drop(t) % (w:real^3)) - x) = &0)}`
;`{(t:real^1) | ~(vector_angle (v - x) (((&1 - drop (t)) % u + drop (t) % w) - (x:real^3)) = pi)}`]OPEN_INTER)
THEN ASM_REWRITE_TAC[INTER;IN_ELIM_THM]
THEN DISCH_TAC
THEN SUBGOAL_THEN` 
((vec 0):real^1) IN {(t:real^1) | ~(   
vector_angle ((v:real^3) - x) (((&1 - drop(t)) % (u:real^3) + drop(t) % (w:real^3)) - x) = &0 )/\ 
~(vector_angle (v - x) (((&1 - drop (t)) % u + drop (t) % w) - (x:real^3)) = pi)}` ASSUME_TAC
THENL(*3*)[

SIMP_TAC[IN_ELIM_THM;drop;VEC_COMPONENT;GSYM(DE_MORGAN_THM)]
THEN REDUCE_ARITH_TAC
THEN REDUCE_VECTOR_TAC
THEN MP_TAC(ISPECL[`((v:real^3)-(x:real^3))`;`((u:real^3)-(x:real^3))`]COLLINEAR_VECTOR_ANGLE)
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A-B=vec 0<=> A=B:real^3`]
THEN ONCE_REWRITE_TAC[GSYM(COLLINEAR_3);]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN ASM_REWRITE_TAC[];(*3*)


POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"G")
THEN DISCH_TAC
THEN REMOVE_THEN "G" MP_TAC
THEN REWRITE_TAC[open_def]
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> MP_TAC(ISPEC`(vec 0):real^1`th))
THEN ASM_REWRITE_TAC[IN_ELIM_THM;dist; VECTOR_ARITH`A-vec 0=A`]
THEN STRIP_TAC
THEN EXISTS_TAC`min ((e:real)/ &2) (&1)`
THEN STRIP_TAC
THENL(*4*)[
POP_ASSUM (fun th->REWRITE_TAC[])
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;(*4*)

STRIP_TAC
THENL(*5*)[
REAL_ARITH_TAC;
GEN_TAC
THEN POP_ASSUM(fun th-> MP_TAC(ISPEC`lift(t:real)`th))
THEN REWRITE_TAC[LIFT_DROP;NORM_LIFT]
THEN DISCH_THEN(LABEL_TAC"G")
THEN STRIP_TAC
THEN REMOVE_THEN "G" MATCH_MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC]]]]]);;




let exist_close_fan=prove(`(!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (w:real^3) (v1:real^3) (w1:real^3).

FAN(x,V,E) /\ {v,w} INTER {v1,w1}={} /\ {v1,w1} IN E /\ {v,w} IN E
==>
?h:real.
(&0 < h)
/\ 
(!y1:real^3 y2:real^3. y1 IN (aff_ge {x} {v, w} INTER ballnorm_fan x) /\ y2 IN (aff_ge {x} {v1, w1} INTER ballnorm_fan x)
==> h  <= dist(y1,y2) ))`,

REPEAT STRIP_TAC 
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (v:real^3)`] 
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w1:real^3)`;
` (v1:real^3)`]
THEN MATCH_MP_TAC( ISPECL [`aff_ge {(x:real^3)} {(v:real^3), (w:real^3)} INTER ballnorm_fan x`; 
 `aff_ge {(x:real^3)} {(v1:real^3), (w1:real^3)} INTER ballnorm_fan x`] SEPARATE_CLOSED_COMPACT)
THEN MP_TAC(ISPECL[`(x:real^3) `;` (v:real^3)`;` (w:real^3)`]closed_aff_ge_ballnorm_fan) 
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(x:real^3) `;` (v1:real^3)`;` (w1:real^3)`]compact_aff_ge_ballnorm_fan) THEN RESA_TAC
THEN FIND_ASSUM(MP_TAC)`FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN;fan7] 
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MP_TAC(ISPECL[`{(v:real^3),(w:real^3)}`;`{(v1:real^3),(w1:real^3)}`]th))
		   THEN ASM_REWRITE_TAC[UNION; IN_ELIM_THM;]
THEN ASM_REWRITE_TAC[AFF_GE_EQ_AFFINE_HULL; SET_RULE`(A INTER C) INTER (B INTER C)= (A INTER B) INTER C`;] 
THEN ASSUME_TAC(AFFINE_SING) 
THEN MP_TAC(ISPEC`{ (x:real^3) }` AFFINE_HULL_EQ )
THEN RESA_TAC 
THEN RESA_TAC 
THEN REWRITE_TAC[ballnorm_fan;INTER; IN_SING; EXTENSION;EMPTY;IN_ELIM_THM;] 
THEN GEN_TAC
THEN STRIP_TAC 
THEN POP_ASSUM MP_TAC 
THEN ASM_REWRITE_TAC[DIST_REFL ] 
THEN REAL_ARITH_TAC);;
let AFF_GT_1_2=prove(`!x v w.
         DISJOINT {x} {v, w}
         ==> aff_gt {x} {v, w} =
             {y | ?t1 t2 t3.
                      &0 < t2 /\
                      &0 < t3 /\
                      t1 + t2 + t3 = &1 /\
                      y = t1 % x + t2 % v + t3 % w}`,
AFF_TAC);;



let linear_aff_fan=prove(`!x:real^3 v:real^3 u:real^3.
linear (\(t:real^2). t$1 %(v-x)+t$2 %(u-x))`,
REPEAT STRIP_TAC
THEN MATCH_MP_TAC LINEAR_COMPOSE_ADD 
THEN STRIP_TAC
THEN MATCH_MP_TAC LINEAR_VMUL_COMPONENT
THEN SIMP_TAC[LINEAR_ID; DIMINDEX_2; ARITH]);;
let linear1_aff_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3.
linear (\(t:real^3). t$1 %(v-x)+t$2 %(u-x)+t$3 %(w-u))`,
REPEAT STRIP_TAC
THEN REPEAT(MATCH_MP_TAC LINEAR_COMPOSE_ADD THEN STRIP_TAC)
THEN MATCH_MP_TAC LINEAR_VMUL_COMPONENT
THEN SIMP_TAC[LINEAR_ID; DIMINDEX_3; ARITH]);;




let linear_inj_fan=prove(`!x:real^3 v:real^3 u:real^3.
~collinear{x,v,u}
==>(!(a:real^2) (b:real^2). (\(t:real^2). t$1 %(v-x)+t$2 %(u-x))(a)=(\(t:real^2). t$1 %(v-x)+t$2 %(u-x))(b) ==>a=b)`,

REPEAT GEN_TAC
THEN DISCH_TAC
THEN ASSUME_TAC(ISPECL[`x:real^3`;` v:real^3`;`u:real^3`]linear_aff_fan)
THEN MP_TAC(ISPEC`(\(t:real^2). t$1 %(v-x)+t$2 %(u-x):real^3)`LINEAR_INJECTIVE_0)
THEN RESP_TAC
THEN REMOVE_ASSUM_TAC
THEN GEN_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(a:real^2)$2= &0 \/ ~(a$2= &0)`)
THENL[
ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[VECTOR_MUL_EQ_0;VECTOR_ARITH`A-B=vec 0<=> B=A`]
THEN MP_TAC(ISPECL[`x:real^3`;` v:real^3`;`u:real^3`]th3)
THEN RESA_TAC
THEN ASM_TAC
THEN SIMP_TAC[ LAMBDA_BETA;CART_EQ; DIMINDEX_2; FORALL_2; VEC_COMPONENT; ARITH];

REWRITE_TAC[VECTOR_ARITH`A+B=vec 0<=>B= --A`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`a$2 % (u - x) = --((a:real^2)$1 % (v - x:real^3)) ==> (inv (a$2)) % a$2 % (u - x) = (inv (a$2)) % (--(a$1 % (v - x)))`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th] 
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(A%B%C=(A*B)%C:real^3)`])
THEN MP_TAC(ISPEC`(a:real^2)$2`REAL_MUL_LINV)
THEN RESA_TAC
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[VECTOR_ARITH`A-B=C%(--(D%(U-B)))<=> A= (&1+C*D)%B+(--C*D)%U:real^3`]
THEN MP_TAC(ISPECL[`x:real^3`;` v:real^3`;`u:real^3`]th3)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[aff; AFFINE_HULL_2;IN_ELIM_THM]
THEN DISCH_THEN(LABEL_TAC"A")
THEN DISCH_TAC
THEN SUBGOAL_THEN `F`ASSUME_TAC
THENL[
REMOVE_THEN "A" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN EXISTS_TAC`(&1 + inv ((a:real^2)$2) * a$1)`
THEN EXISTS_TAC`(--inv ((a:real^2)$2) * a$1)`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC;
ASM_MESON_TAC[]]]);;




let origin_point_not_in_convex_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) (w:real^3).
FAN(x,V,E) /\ {v,u} IN E/\ {u,w} IN E /\ ~(coplanar{x,v,u,w})
==>
~(x IN convex hull{v,u,w})`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[CONVEX_HULL_3; IN_ELIM_THM; coplanar]
THEN STRIP_TAC
THEN EXISTS_TAC`v:real^3`
THEN EXISTS_TAC`u:real^3`
THEN EXISTS_TAC`w:real^3`
THEN SUBGOAL_THEN`(x:real^3)IN affine hull {v,u,w:real^3}` ASSUME_TAC
THENL(*3*)[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`u':real`
THEN EXISTS_TAC`v':real`
THEN EXISTS_TAC`w':real`
THEN ASM_MESON_TAC[];

SUBGOAL_THEN`(v:real^3)IN affine hull {v,u,w:real^3}` ASSUME_TAC
THENL(*4*)[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&0`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;

SUBGOAL_THEN`(u:real^3)IN affine hull {v,u,w:real^3}` ASSUME_TAC
THENL(*5*)[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC`&0`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;
SUBGOAL_THEN`(w:real^3)IN affine hull {v,u,w:real^3}` ASSUME_TAC
THENL(*6*)[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;
ASM_TAC
THEN SET_TAC[]]]]]);;


let separate_point_convex_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) (w:real^3).
FAN(x,V,E) /\ {v,u} IN E/\ {u,w} IN E /\ ~(coplanar{x,v,u,w})
==>
?(h:real). &0< h /\ (!(y:real^3).  y IN convex hull{v,u,w} ==> h < norm(y-x))`,

REPEAT STRIP_TAC 
THEN SUBGOAL_THEN `FINITE {(v:real^3),(u:real^3),(w:real^3)}` ASSUME_TAC
THENL(*1*)[
MP_TAC(ISPECL[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`] remark1_fan)
THEN RESA_TAC 
THEN MP_TAC(ISPECL[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
`(u:real^3)`] remark1_fan)
THEN RESA_TAC 
THEN MP_TAC(ISPECL[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (w:real^3)`] remark1_fan)
THEN RESA_TAC 
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(v:real^3) IN (V:real^3->bool) /\ (u:real^3) IN (V:real^3->bool) /\ (w:real^3) IN (V:real^3->bool)==> {v,u,w} SUBSET V`)
THEN ASM_REWRITE_TAC[]
THEN DISCH_TAC
THEN MATCH_MP_TAC(ISPECL[`{(v:real^3),(u:real^3),(w:real^3)}`;`V:real^3->bool`]FINITE_SUBSET)
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REWRITE_TAC[FAN;fan1]
THEN SET_TAC[];

MP_TAC(ISPEC`{(v:real^3),(u:real^3),(w:real^3)}`FINITE_IMP_COMPACT_CONVEX_HULL)
THEN POP_ASSUM (fun th-> REWRITE_TAC[th;COMPACT_EQ_BOUNDED_CLOSED])
THEN DISCH_TAC
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (v:real^3) `;`(u:real^3) `;`(w:real^3)`]origin_point_not_in_convex_fan)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`convex hull {(v:real^3),(u:real^3),(w:real^3)}`;`x:real^3`]SEPARATE_POINT_CLOSED)
THEN RESA_TAC
THEN EXISTS_TAC`d:real/ &2`
THEN STRIP_TAC
THENL[
ASM_TAC THEN REAL_ARITH_TAC;

ONCE_REWRITE_TAC[NORM_SUB]
THEN GEN_TAC
THEN POP_ASSUM (fun th ->MP_TAC(ISPEC`y:real^3`th))
THEN REWRITE_TAC[dist;]
THEN DISCH_THEN(LABEL_TAC"A")
THEN DISCH_TAC 
THEN REMOVE_THEN "A" MP_TAC
THEN POP_ASSUM (fun th ->REWRITE_TAC[th])
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC]]);;



let expansion_convex_fan=prove(`!(v:real^3) (u:real^3) (w:real^3) (t:real) s:real.
&0 <= t /\ t<= &1
/\ &0 <=s /\ s <= &1 
==> (&1-s)%v+s%((&1-t)%u+ t%w) IN convex hull{v,u,w}`,

REWRITE_TAC[REAL_ARITH`A<= &1 <=> &0<= &1 -A`]
THEN REPEAT STRIP_TAC
THEN REWRITE_TAC[CONVEX_HULL_3; IN_ELIM_THM;]
THEN EXISTS_TAC`&1 - (s:real)`
THEN EXISTS_TAC`(s:real)*(&1 - (t:real))`
THEN EXISTS_TAC`(s:real)*(t:real)`
THEN ASM_REWRITE_TAC[VECTOR_ARITH`s%((&1-t)%u+ t%w)= (s*(&1-t))%u+ (s*t)%w:real^3`]
THEN STRIP_TAC
THENL[MATCH_MP_TAC REAL_LE_MUL
THEN ASM_TAC
THEN REAL_ARITH_TAC;
STRIP_TAC
THENL[MATCH_MP_TAC REAL_LE_MUL
THEN ASM_TAC
THEN REAL_ARITH_TAC;
REAL_ARITH_TAC]]);;

let expansion1_convex_fan=prove(`!(v:real^3) (u:real^3) s:real.
&0 <=s /\ s <= &1 
==> (&1-s)%v+s%u IN convex hull{v,u}`,
REPEAT STRIP_TAC
THEN MP_TAC(ISPECL[`(v:real^3)`;` (u:real^3)`;` (u:real^3)`;` &0`;`s:real`]expansion_convex_fan)
THEN ASM_REWRITE_TAC[SET_RULE`{A,B,B}={A,B}`]
THEN REDUCE_ARITH_TAC
THEN REDUCE_VECTOR_TAC
THEN DISCH_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN REAL_ARITH_TAC);;
let norm_origin_fan=prove(`!x:real^3.
 (\(y:real^3). lift(norm(y-x))) continuous_on (:real^3) `,
GEN_TAC
THEN MP_TAC(ISPECL[`(\(y:real^3). y-(x:real^3))`;`(\(y:real^3). lift(norm(y)))`;`(:real^3)`]CONTINUOUS_ON_COMPOSE)
THEN REWRITE_TAC[o_DEF]
THEN DISCH_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN REWRITE_TAC[CONTINUOUS_ON_LIFT_NORM;GSYM(o_DEF)]
THEN MATCH_MP_TAC CONTINUOUS_ON_SUB 
THEN SIMP_TAC[CONTINUOUS_ON_CONST;CONTINUOUS_ON_ID]);;


let origin_point_not1_in_convex_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3).
FAN(x,V,E) /\ {v,u} IN E
==>
~(x IN convex hull{v,u})`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ISPECL[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`] remark1_fan)
THEN RESA_TAC
THEN FIND_ASSUM MP_TAC`~(u IN aff {x,v:real^3})`
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[CONVEX_HULL_2; IN_ELIM_THM;]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(v':real= &0) \/ ~(v':real= &0)`)
THENL[ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN DISCH_TAC
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN ASM_MESON_TAC[];

POP_ASSUM MP_TAC
THEN REPEAT REMOVE_ASSUM_TAC
THEN REWRITE_TAC[VECTOR_ARITH`A=B+C<=>C= A-B:real^3`;REAL_ARITH`a+b= &1<=> a= &1 -b:real`]
THEN REPEAT DISCH_TAC
THEN MP_TAC(SET_RULE`v'% u =x- u' % v ==> (inv (v')) % v' % u = (inv (v')) % (x-u' % v:real^3)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th] 
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`;VECTOR_ARITH`(A%(B-(&1-U)%C)=(A%B)+(A*U-A)%C:real^3)`])
THEN MP_TAC(ISPEC`(v':real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`inv(v':real)`
THEN EXISTS_TAC`&1-inv(v':real)`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC]);;





let inequality1_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) (w:real^3) d:real.
FAN(x,V,E) /\ {v,u} IN E/\ {u,w} IN E /\ ~(coplanar{x,v,u,w}) /\ &0 < d
==>
?(h:real). &0 <h /\  h<= &1
/\ (!t:real. &0 <= t /\  t<h
==> (!(s:real). &0 <= s /\ s <= &1 
==> s * inv(norm((&1-s)%v+s%((&1-t)%u+ t%w)-x))* norm(u-((&1-t)%u+ t%w))< d  ))`,
REPEAT STRIP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`u-((&1-t)%u+ t%w)=t%(u-w):real^3`;NORM_MUL]
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (v:real^3)`;` (u:real^3)`;` (w:real^3)`]separate_point_convex_fan)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"A")
THEN EXISTS_TAC`min ((h:real)*(d:real) *inv(norm(u-w:real^3))) (&1)`
THEN MP_TAC(ISPECL[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] remark1_fan)
THEN RESA_TAC 
THEN  MP_TAC(ISPECL[`u:real^3`;`w:real^3`]imp_norm_gl_zero_fan)
THEN RESA_TAC 
THEN MP_TAC(ISPECL[`(h:real)*(d:real)`;`inv(norm(u-w:real^3))`] REAL_LT_MUL)
THEN RESA_TAC
THEN STRIP_TAC
THENL[
MP_TAC(ISPECL[`h:real`;`d:real`] REAL_LT_MUL)
THEN RESA_TAC
THEN ASM_TAC
THEN REAL_ARITH_TAC;
STRIP_TAC
THENL[

REAL_ARITH_TAC;
REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0 <(h:real)==> ~(h= &0)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(h:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`(t:real)< min ((h:real) * (d:real) * inv (norm (u - w:real^3))) (&1)/\ min (h * d * inv (norm (u - w))) (&1)<= &1 ==>t<= &1/\ (t:real)< (h * d * inv (norm (u - w:real^3)))`)
THEN ASM_REWRITE_TAC[REAL_ARITH`min (h * d * inv (norm (u - w))) (&1)<= &1`]
THEN STRIP_TAC
THEN MP_TAC(ISPECL[`(v:real^3)`;` (u:real^3) `;`(w:real^3) `;`(t:real) `;`s:real`]expansion_convex_fan)
THEN RESA_TAC
THEN REMOVE_THEN "A"(fun th -> MP_TAC(ISPEC`(&1-s)%v+s%((&1-t)%u+ t%w):real^3`th))
THEN RESA_TAC
THEN MP_TAC(ISPECL[`h:real`;`norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3)`]REAL_LT_INV2)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0 <h /\ (h:real)< norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3)==> &0 <= norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3)`REAL_LE_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`t:real`REAL_ABS_REFL)
THEN RESP_TAC
THEN MP_TAC(ISPECL[`inv(norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3))`;
`inv(h:real)`;`t:real`;`(h * d * inv (norm (u - w:real^3)))`;
]REAL_LT_MUL2)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`u:real^3`;`w:real^3`]imp_norm_not_zero_fan)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(norm(u-w:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`u-w:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(&0 =norm(u-w:real^3)) /\ &0 <= norm(u-w:real^3)==> &0 <norm(u-w:real^3)`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`inv (norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3)) * (t:real)`;`inv (h:real) * h * d * inv (norm (u - w:real^3))`;`norm (u - w:real^3)`]REAL_LT_RMUL)
THEN ASM_REWRITE_TAC[REAL_ARITH`(A*B*C*D)*E=(A*B)*C*(D*E)`]
THEN REDUCE_ARITH_TAC
THEN STRIP_TAC
THEN MP_TAC(ISPECL[`inv (norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3))`;`(t:real)`;] REAL_LE_MUL)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`inv (norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3))*(t:real)`;`norm(u-w:real^3)`] REAL_LE_MUL)
THEN RESA_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(s= &1) \/ ~((s:real)= &1)`)
THENL[
POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN REDUCE_ARITH_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`A*B*C=(A*B)*C:real`;VECTOR_ARITH`(&1 - s) % v + s % ((&1 - t) % u + t % w) - x=((&1 - s) % v + s % ((&1 - t) % u + t % w))- x :real^3`];
MP_TAC(REAL_ARITH`~(s= &1) /\ s<= &1==> (s:real)< &1`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`s:real`;`&1`;`(inv (norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3))*(t:real)) * norm(u-w:real^3)`;
`d:real`]REAL_LT_MUL2)
THEN REDUCE_ARITH_TAC
THEN RESA_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`A*d*B*C=A*(d*B)*C:real`;VECTOR_ARITH`(&1 - s) % v + s % ((&1 - t) % u + t % w) - x=((&1 - s) % v + s % ((&1 - t) % u + t % w))- x :real^3`]]]]);;



let bounded_convex_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3).
FAN(x,V,E) /\ {v,u} IN E
==> ?(h:real). &0< h /\ (!(y:real^3).  y IN convex hull{v,u} ==> norm(y-x)<h )`,
REPEAT STRIP_TAC
THEN SUBGOAL_THEN `FINITE {(v:real^3),(u:real^3)}` ASSUME_TAC
THENL(*1*)[SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY;
                 IN_INSERT; NOT_IN_EMPTY];

MP_TAC(ISPEC`{(v:real^3),(u:real^3)}`FINITE_IMP_COMPACT_CONVEX_HULL)
THEN POP_ASSUM (fun th-> REWRITE_TAC[th;])
THEN DISCH_TAC
THEN ASSUME_TAC(ISPEC`x:real^3` norm_origin_fan )
THEN ASSUME_TAC(SET_RULE`convex hull {(v:real^3), u} SUBSET (:real^3)`)
THEN MP_TAC(ISPECL[`(\(y:real^3). lift(norm(y-x:real^3)))`;`(:real^3)`;`convex hull {(v:real^3),(u:real^3)}`]CONTINUOUS_ON_SUBSET)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(\(y:real^3). lift(norm(y-x:real^3)))`;`convex hull {(v:real^3),(u:real^3)}`]COMPACT_CONTINUOUS_IMAGE)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED;bounded;IMAGE;IN_ELIM_THM]
THEN ONCE_REWRITE_TAC[TAUT`A/\B<=>B/\A`;]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"A")
THEN SUBGOAL_THEN `(?x':real^3. x' IN convex hull {v, u} /\
        lift (norm (v - x:real^3)) = lift (norm (x' - x)))`ASSUME_TAC
THENL[
EXISTS_TAC`v:real^3`
THEN SIMP_TAC[CONVEX_HULL_2; IN_ELIM_THM;]
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC`&0`
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC;

SUBGOAL_THEN `&0< (a:real)` ASSUME_TAC
THENL[
REMOVE_THEN "A" (fun th-> MP_TAC(ISPEC`lift (norm (v - (x:real^3)))`th))
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[NORM_LIFT;REAL_ABS_NORM ]
THEN MP_TAC(ISPECL[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`] remark1_fan)
THEN RESA_TAC 
THEN MP_TAC(ISPECL[`v:real^3`;`x:real^3`]imp_norm_not_zero_fan)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`v-x:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(&0 =norm(v-x:real^3)) /\ &0 <= norm(v-x:real^3)==> &0 <norm(v-x:real^3)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;

EXISTS_TAC`&2 * a:real`
THEN STRIP_TAC
THENL[

REPEAT STRIP_TAC
THEN SUBGOAL_THEN `(?x':real^3. x' IN convex hull {v, u} /\
        lift (norm (y - x:real^3)) = lift (norm (x' - x)))`ASSUME_TAC
THENL[
EXISTS_TAC`y:real^3`
THEN ASM_SIMP_TAC[];

REMOVE_THEN "A"(fun th-> MP_TAC(ISPEC`lift (norm (y - x:real^3))`th)) 
THEN ASM_REWRITE_TAC[NORM_LIFT;REAL_ABS_NORM ]
THEN ASM_TAC
THEN REAL_ARITH_TAC];

ASM_TAC
THEN REAL_ARITH_TAC]]]]);;


let REAL_ABS_SUB_NORM = prove
 (`!x y. abs(norm(x) - norm(y)) <= norm(x - y)`,
  REWRITE_TAC[REAL_ARITH `abs(x - y) <= a <=> x <= y + a /\ y <= x + a`] THEN
  MESON_TAC[NORM_TRIANGLE_SUB; NORM_SUB]);;


let inequaility2_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) (w:real^3) d:real.
FAN(x,V,E) /\ {v,u} IN E/\ {u,w} IN E /\ ~(coplanar{x,v,u,w}) /\ &0 < d
==>
?(h:real). &0 <h /\  h<= &1
/\ (!t:real. &0 <= t /\  t<h
==> (!(s:real). &0 <= s /\ s <= &1 
==> (norm(inv(norm((&1-s)%v+s%u-x))%((&1-s)%v+s%u-x) - inv(norm((&1-s)%v+s%((&1-t)%u+ t%w)-x))% ((&1-s)%v+s%u-x)))< d  ))`,

REWRITE_TAC[VECTOR_ARITH`A%B-C%B=(A-C)%B`;NORM_MUL]
THEN REPEAT STRIP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`u-((&1-t)%u+ t%w)=t%(u-w):real^3`;NORM_MUL]
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (v:real^3)`;` (u:real^3)`;` (w:real^3)`]separate_point_convex_fan)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"A")
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (v:real^3) `;`(u:real^3) `;`(w:real^3)`]origin_point_not_in_convex_fan)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"B")
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;`(u:real^3)`]bounded_convex_fan)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"C")
THEN EXISTS_TAC`min (h*h*(inv (h'))* inv(norm(u-w:real^3))*d:real) (&1)`
THEN MP_TAC(ISPECL[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] remark1_fan)
THEN RESA_TAC 
THEN MP_TAC(ISPECL[`u:real^3`;`w:real^3`]imp_norm_not_zero_fan)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`u-w:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(&0 =norm(u-w:real^3)) /\ &0 <= norm(u-w:real^3)==> &0 <norm(u-w:real^3)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(norm(u-w:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0 <(h:real)==> ~(h= &0)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(h:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0 <(h':real)==> ~(h'= &0)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(h':real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`norm (u-w:real^3)`REAL_LE_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`norm (u-w:real^3)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(h:real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0 <inv (h:real)==> &0 <= inv(h)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(h':real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0 <inv (h':real)==> &0 <= inv(h')`)
THEN RESA_TAC
THEN STRIP_TAC
THENL(*1*)[
MP_TAC(ISPECL[`(h:real)`;`inv (h':real) `]REAL_LT_MUL)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(h:real)`;`(h:real)*inv (h':real) `]REAL_LT_MUL)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(h:real)*(h:real)*inv (h':real)`;`inv(norm(u-w:real^3))`]REAL_LT_MUL)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(h:real)*(h:real)*inv (h':real) *inv(norm(u-w:real^3))`;`d:real`]REAL_LT_MUL)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;(*1*)


STRIP_TAC
THENL(*2*)[

REAL_ARITH_TAC;(*2*)

REPEAT STRIP_TAC
THEN MP_TAC(ISPECL[`(v:real^3)`;` (u:real^3)`;` s:real`]expansion1_convex_fan)
THEN RESA_TAC
THEN REMOVE_THEN "C"(fun th->MP_TAC(ISPEC`(&1 - s) % v + s % u:real^3`th))
THEN REWRITE_TAC[VECTOR_ARITH`(A+B)-C=A+B-C:real^3`]
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;`(u:real^3)`]
origin_point_not1_in_convex_fan)
THEN RESA_TAC
THEN SUBGOAL_THEN`~(x=(&1-s)%v+s%u:real^3)` ASSUME_TAC
THENL(*3*)[
POP_ASSUM MP_TAC
THEN MATCH_MP_TAC MONO_NOT
THEN RESA_TAC;(*3*)

MP_TAC(ISPECL[`(&1-s)%v+s%u:real^3`;`x:real^3`]imp_norm_not_zero_fan)
THEN REWRITE_TAC[VECTOR_ARITH`(A+B)-C=A+B-C:real^3`]
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`(&1-s)%v+s%u-x:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(norm((&1-s)%v+s%u-x:real^3)= &0) /\ &0 <= norm((&1-s)%v+s%u-x:real^3)==> &0 <norm((&1-s)%v+s%u-x:real^3)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(norm((&1-s)%v+s%u-x:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(norm((&1-s)%v+s%u-x:real^3))`REAL_LE_INV)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`(t:real)< min (h * h * inv h' * inv (norm (u - w)) * d) (&1)/\ min (h * h * inv h' * inv (norm (u - w)) * d) (&1)<= &1 ==> t<= &1/\ (t:real)< (h * h * inv h' * inv (norm (u - w:real^3)) * d) `)
THEN ASM_REWRITE_TAC[REAL_ARITH`min (h * h * inv h' * inv (norm (u - w:real^3)) * d) (&1)<= &1`]
THEN STRIP_TAC
THEN MP_TAC(ISPECL[`norm ((&1 - s) % v + s % u - x:real^3) `;
`(h':real)`;`t:real`;`(h * h * inv h' * inv (norm (u - w:real^3)) * d)`;]REAL_LT_MUL2)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(v:real^3)`;` (u:real^3) `;`(w:real^3) `;`(&0:real) `;`s:real`]expansion_convex_fan)
THEN REDUCE_ARITH_TAC
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= &0/\ &0<= &1`]
THEN DISCH_TAC
THEN USE_THEN "A" (fun th-> MP_TAC(ISPEC`((&1 - s) % v + s % u:real^3)`th))
THEN POP_ASSUM(fun th-> REWRITE_TAC[th; VECTOR_ARITH`(A+B)-C=A+B-C:real^3`])
THEN DISCH_TAC
THEN MP_TAC(ISPECL[`h:real`;`norm ((&1 - s) % v + s % u-x:real^3)`]REAL_LT_INV2)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`norm((&1-s)%v+s%u-x:real^3)`;`t:real`]REAL_LE_MUL)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`inv(norm ((&1 - s) % v + s % u - x:real^3))`;`inv(h:real)`;`norm ((&1 - s) % v + s % u - x:real^3)*(t:real)`;`(h':real)*(h * h * inv h' * inv (norm (u - w:real^3)) * d)`;]REAL_LT_MUL2)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(v:real^3)`;` (u:real^3) `;`(w:real^3) `;`(t:real) `;`s:real`]expansion_convex_fan)
THEN ASM_REWRITE_TAC[]
THEN DISCH_TAC
THEN USE_THEN "A" (fun th-> MP_TAC(ISPEC`(&1 - s) % v + s % ((&1 - t) % u + t % w):real^3`th))
THEN POP_ASSUM(fun th-> REWRITE_TAC[th; VECTOR_ARITH`(A+B)-C=A+B-C:real^3`])
THEN DISCH_TAC
THEN MP_TAC(ISPECL[`h:real`;`norm ((&1 - s) % v + s % ((&1 - t) % u + t % w)-x:real^3)`]REAL_LT_INV2)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`inv(norm((&1-s)%v+s%u-x:real^3))`;`norm((&1-s)%v+s%u-x:real^3)*(t:real)`]REAL_LE_MUL)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`((&1 - s) % v + s % ((&1 - t) % u + t % w)-x:real^3)`NORM_POS_LE)
THEN MP_TAC(ISPEC`norm ((&1 - s) % v + s % ((&1 - t) % u + t % w)-x:real^3)`REAL_LE_INV)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`inv(norm ((&1 - s) % v + s % ((&1 - t) % u + t % w)-x:real^3))`;`inv(h:real)`;`inv(norm ((&1 - s) % v + s % u - x:real^3)) * norm ((&1 - s) % v + s % u - x:real^3)*(t:real)`;`inv(h:real)*(h':real)*(h * h * inv h' * inv (norm (u - w:real^3)) * d)`;]REAL_LT_MUL2)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`norm (u - w:real^3)`;
`inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x:real^3)) *
      inv (norm ((&1 - s) % v + s % u - x)) *
      norm ((&1 - s) % v + s % u - x) *
      t`;`inv h * inv h * h' * h * h * inv h' * inv (norm (u - w:real^3)) * d`;]REAL_LT_LMUL)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[REAL_ARITH`A*B*C*D*E=(B*C)*E*A*D`]
THEN ONCE_REWRITE_TAC[REAL_ARITH`A*B*C*D*E=(A*B)*E*(C*D)`]
THEN ONCE_REWRITE_TAC[REAL_ARITH`A*B*C*D*E=(A*B)*E*(C*D)`]
THEN ONCE_REWRITE_TAC[REAL_ARITH`A*B*C*D=(A*B)*C*D`]
THEN ONCE_REWRITE_TAC[REAL_ARITH`A*B*C*D*E=(A*B)*(C*D)*E`]
THEN ONCE_REWRITE_TAC[REAL_ARITH`(A*B*C)*D*E=A*B*(C*D)*E`]
THEN ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B*C)*D=A*B*(C*D)`]
THEN REDUCE_ARITH_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`((A*B)*C*D)*E=(A*C)*(B*D)*E`]
THEN REDUCE_ARITH_TAC
THEN REWRITE_TAC[REAL_ARITH`((A*B)*C)*D*E=(A*B)*(C*D)*E`]
THEN ASSUME_TAC(ISPECL[`u:real^3`;`w:real^3`]NORM_SUB)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN MP_TAC(ISPECL[`t:real`;`w-u:real^3`]NORM_MUL)
THEN MP_TAC(ISPEC`t:real`REAL_ABS_REFL)
THEN RESP_TAC
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN ASM_REWRITE_TAC[REAL_ARITH`(A*B)*C*D=C*A*(B*D)`]
THEN REDUCE_ARITH_TAC
THEN DISCH_THEN(LABEL_TAC"BA")
THEN MP_TAC(ISPECL[`(v:real^3)`;` (u:real^3) `;`(w:real^3) `;`(t:real) `;`s:real`]expansion_convex_fan)
THEN RESA_TAC
THEN SUBGOAL_THEN`~(x=(&1 - s) % v + s % ((&1 - t) % u + t % w):real^3)` ASSUME_TAC
THENL(*4*)[
POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"MA")
THEN STRIP_TAC
THEN REMOVE_THEN"MA" MP_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN ASM_REWRITE_TAC[];(*4*)

MP_TAC(ISPECL[`(&1 - s) % v + s % ((&1 - t) % u + t % w):real^3`;`x:real^3`]imp_norm_not_zero_fan)
THEN REWRITE_TAC[VECTOR_ARITH`(A+B)-C=A+B-C:real^3`]
THEN RESA_TAC
THEN MP_TAC(ISPECL[`norm((&1 - s) % v + s % u-x:real^3)`;`norm((&1 - s) % v + s % ((&1 - t) % u + t % w)-x:real^3)`]REAL_SUB_INV)
THEN RESP_TAC
THEN ASM_REWRITE_TAC[REAL_ABS_ABS;REAL_ABS_DIV;REAL_ABS_MUL;REAL_ABS_NORM;real_div;REAL_INV_MUL;
REAL_ARITH`(A*B*C)*D=A*C*(B*D)`]
THEN REDUCE_ARITH_TAC
THEN MP_TAC(ISPECL[`((&1 - s) % v + s % ((&1 - t) % u + t % w)-x:real^3)`;`((&1 - s) % v + s % u-x:real^3)`]REAL_ABS_SUB_NORM)
THEN REWRITE_TAC[VECTOR_ARITH`
((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)-((&1 - s) % v + s % u - x)=s%(t)%(w-u) :real^3`]
THEN ONCE_REWRITE_TAC[NORM_MUL]
THEN MP_TAC(ISPEC`s:real`REAL_ABS_REFL)
THEN RESP_TAC
THEN DISCH_TAC
THEN MP_TAC(ISPECL[`abs
 (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x:real^3) -
  norm ((&1 - s) % v + s % u - x))`;`s * norm (t % (w - u):real^3)`;`inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x:real^3))`]REAL_LE_RMUL)
THEN RESA_TAC
THEN REMOVE_THEN "BA" MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(s= &1)\/ ~((s:real) = &1)`)
THENL(*5*)[
ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC;(*5*)

MP_TAC(REAL_ARITH`~(s= &1) /\ s<= &1==> (s:real)< &1`)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`(t % (w - u):real^3)`NORM_POS_LE)
THEN MP_TAC(ISPECL[`norm (t % (w - u):real^3)`;`inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x:real^3))`]REAL_LE_MUL)
THEN RESA_TAC
THEN REPEAT STRIP_TAC
THEN MP_TAC(ISPECL[`s:real`;`&1`;`norm (t % (w - u)) *
     inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x:real^3))`;
`d:real`]REAL_LT_MUL2)
THEN REDUCE_ARITH_TAC
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC]]]]]);;




let exists_point_small_edges_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) (w:real^3) d:real.
FAN(x,V,E) /\ {v,u} IN E/\ {u,w} IN E /\ ~(coplanar{x,v,u,w}) /\ &0 < d
==>
?(h:real). &0 <h /\  h<= &1
/\ (!t:real. &0 <= t /\  t<h
==> (!(s:real). &0 <= s /\ s <= &1 
==> norm(inv(norm((&1-s)%v+s%u-x))%((&1-s)%v+s%u-x) - inv(norm((&1-s)%v+s%((&1-t)%u+ t%w)-x))% ((&1-s)%v+s%((&1-t)%u+ t%w)-x))< d  ))`,

REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0< (d:real)==> &0< d/ &2`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (v:real^3) `;`(u:real^3) `;`(w:real^3)`]origin_point_not_in_convex_fan)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (v:real^3) `;`(u:real^3) `;`(w:real^3)`;` (d:real)/ &2`]inequaility2_fan)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"A")
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;`(E:(real^3->bool)->bool) `;`(v:real^3)`;` (u:real^3)`;` (w:real^3)`;` (d:real)/ &2`]inequality1_fan)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"B")
THEN EXISTS_TAC`min (h:real) (h':real)`
THEN STRIP_TAC
THENL[ASM_TAC
THEN REAL_ARITH_TAC;

STRIP_TAC
THENL[
ASM_TAC
THEN REAL_ARITH_TAC;

REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`t<  min (h:real) (h':real)==> t< (h:real) /\ t< (h':real)`)
THEN RESA_TAC
THEN REMOVE_THEN"B" (fun th-> MP_TAC(ISPEC`t:real`th))
THEN RESA_TAC
THEN POP_ASSUM  (fun th-> MP_TAC(ISPEC`s:real`th))
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"B")
THEN REMOVE_THEN"A" (fun th-> MP_TAC(ISPEC`t:real`th))
THEN RESA_TAC
THEN POP_ASSUM  (fun th-> MP_TAC(ISPEC`s:real`th))
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"A")
THEN ASSUME_TAC(ISPEC`((&1 - s) % v + s % ((&1 - t) % u + t % w)-x:real^3)`NORM_POS_LE)
THEN MP_TAC(ISPEC`norm ((&1 - s) % v + s % ((&1 - t) % u + t % w)-x:real^3)`REAL_LE_INV)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`s *
      inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x:real^3)):real`;`(u - ((&1 - t) % u + t % w)):real^3`]NORM_MUL)
THEN REWRITE_TAC[REAL_ABS_MUL]
THEN MP_TAC(ISPEC`s:real`REAL_ABS_REFL)
THEN RESP_TAC
THEN MP_TAC(ISPEC`inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x:real^3)):real`REAL_ABS_REFL)
THEN RESP_TAC
THEN REWRITE_TAC[REAL_ARITH`(A*B)*C=A*B*C`]
THEN DISCH_TAC
THEN REMOVE_THEN"B"MP_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN REWRITE_TAC[VECTOR_ARITH`(s * inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x))) %
  (u - ((&1 - t) % u + t % w))= inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) % ((&1 - s) % v + s % u - x)- inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
      ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x):real^3`]

THEN DISCH_TAC
THEN MP_TAC(ISPECL[`norm(inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) % ((&1 - s) % v + s % u - x)- inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
      ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x):real^3)`;`(d:real)/ &2`;
`norm
      (inv (norm ((&1 - s) % v + s % u - x)) % ((&1 - s) % v + s % u - x) -
       inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
       ((&1 - s) % v + s % u - x):real^3)`;`(d:real)/ &2`;
]REAL_LT_ADD2)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`d / &2 + d/ &2 = d`]
THEN MP_TAC(ISPECL[`(inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) % ((&1 - s) % v + s % u - x)- inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
      ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x):real^3)`;
`(inv (norm ((&1 - s) % v + s % u - x)) % ((&1 - s) % v + s % u - x) -
       inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
       ((&1 - s) % v + s % u - x):real^3)`]NORM_TRIANGLE)
THEN REWRITE_TAC[VECTOR_ARITH`(inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) % ((&1 - s) % v + s % u - x)- inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
      ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x):real^3)+
(inv (norm ((&1 - s) % v + s % u - x)) % ((&1 - s) % v + s % u - x) -
       inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
       ((&1 - s) % v + s % u - x):real^3)=(inv (norm ((&1 - s) % v + s % u - x)) % ((&1 - s) % v + s % u - x) -
  inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
  ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) `]
THEN REAL_ARITH_TAC]]);;




let same_projective_sphere_ge_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) (w:real^3) (t:real) (y1:real^3).
FAN(x,V,E) /\ {v,u} IN E/\ {u,w} IN E /\ ~collinear {x, v, (&1 - t) % u + t % w}
/\ ~(y1=x)
/\ (y1 IN aff_ge {x} {v, (&1-t)%u+(t:real)% (w:real^3)} INTER ballnorm_fan x)

==> ?s:real. &0 <= s/\ s<= &1
/\ y1=inv(norm((&1 - s) % v + s % ((&1 - t) % u + t % w)-x))%((&1 - s) % v + s % ((&1 - t) % u + t % w)-x)+x`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"A")
THEN MP_TAC(ISPECL[`(x:real^3)` ;` (v:real^3)`;`(&1 - t) % (u:real^3) + t % (w:real^3)`;]th3)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(x:real^3)` ;` (v:real^3)`;`(&1 - t) % (u:real^3) + t % (w:real^3)`;]AFF_GE_1_2)
THEN RESA_TAC
THEN REMOVE_THEN"A" MP_TAC
THEN ASM_REWRITE_TAC[INTER;IN_ELIM_THM;dist;ballnorm_fan]
THEN STRIP_TAC
THEN SUBGOAL_THEN`~(&1 - t1:real= &0)/\ &0 <= &1-t1` ASSUME_TAC
THENL[
STRIP_TAC
THENL[

FIND_ASSUM MP_TAC`t1+t2+t3= &1:real`
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN REWRITE_TAC[REAL_ARITH`(A+B)-A=B`]
THEN STRIP_TAC
THEN MP_TAC(REAL_ARITH`(&0 <= t2) /\ (&0 <= t3) /\(t2+t3= &0)==> t2= &0 /\ t3= &0`)
THEN RESA_TAC
THEN FIND_ASSUM MP_TAC`y1 = t1 % x + t2 % v + t3 % ((&1 - t) % u + t % w)
:real^3`
THEN FIND_ASSUM MP_TAC`t1+t2+t3= &1:real`
THEN POP_ASSUM (fun th-> REWRITE_TAC[(th)])
THEN POP_ASSUM (fun th-> REWRITE_TAC[(th)])
THEN REDUCE_ARITH_TAC
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[(th)])
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[];
REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ISPECL[`t2:real`;`t3:real`]REAL_LT_ADD)
THEN REAL_ARITH_TAC];

MP_TAC(REAL_ARITH`~(&1 - t1= &0)/\ &0 <= &1-t1==>  &0< (&1 -t1:real)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(&1 -t1:real)`REAL_LE_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(&1-t1:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`inv ((&1 - t1:real))`;`(t2:real)`]REAL_LE_MUL)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`inv ((&1 - t1:real))`;`(t3:real)`]REAL_LE_MUL)
THEN RESA_TAC
THEN FIND_ASSUM MP_TAC `t1+t2+t3= &1:real`
THEN REWRITE_TAC[REAL_ARITH`A+B+C=D<=>B+C=D-A`]
THEN DISCH_TAC
THEN MP_TAC(SET_RULE`t2+t3 = &1-t1 ==> (inv (&1-t1))*(t2+t3) = (inv (&1-t1))* (&1-t1:real)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C)=D<=>D- A*C=A*B`]
THEN REDUCE_ARITH_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`inv ((&1 - t1:real))*(t3:real)`
THEN STRIP_TAC
THENL[
ASM_TAC
THEN REAL_ARITH_TAC;
STRIP_TAC
THENL[
ASM_TAC
THEN REAL_ARITH_TAC;
ASM_REWRITE_TAC[VECTOR_ARITH`(inv (&1 - t1) * t2) % v +
  (inv (&1 - t1) * t3) % ((&1 - t) % u + t % w)-x= inv (&1 - t1) %( t2 % v + t3 % ((&1 - t) % u + t % w))-x:real^3`]
THEN FIND_ASSUM MP_TAC `norm(x-y1:real^3)= &1`
THEN FIND_ASSUM MP_TAC `y1 = t1 % x + t2 % v + t3 % ((&1 - t) % u + t % w):real^3`
THEN DISCH_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[NORM_SUB]
THEN REWRITE_TAC[VECTOR_ARITH`(B%A+C+D)-A=C+D-(&1-B)%A`]
THEN DISCH_TAC
THEN MP_TAC(SET_RULE`norm (t2 % v + t3 % ((&1 - t) % u + t % w) - (&1 - t1) % x:real^3) = &1 ==> (inv (&1-t1))*(norm (t2 % v + t3 % ((&1 - t) % u + t % w) - (&1 - t1) % x) ) = (inv (&1-t1))* (&1):real`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN MP_TAC(ISPECL[`inv(&1-t1:real)`;`t2 % v + t3 % ((&1 - t) % u + t % w) - (&1 - t1) % x:real^3`]NORM_MUL)
THEN MP_TAC(ISPEC`inv(&1-t1:real)`REAL_ABS_REFL)
THEN RESP_TAC
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%(B+C-E%D)=A%(B+C)-(A*E)%D`]
THEN REDUCE_VECTOR_TAC
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[(th)])
THEN ASM_REWRITE_TAC[REAL_INV_INV;VECTOR_ARITH`(&1-A)%(B%C-D)+D= A%D+(B*(&1-A))%C:real^3`]
THEN REDUCE_VECTOR_TAC]]]);;




let same_projective_sphere_gt_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) (w:real^3) (t:real) (y1:real^3).
FAN(x,V,E) /\ {v,u} IN E/\ {u,w} IN E /\ ~collinear {x, v, (&1 - t) % u + t % w}
/\ (y1 IN aff_gt {x} {v, (&1-t)%u+(t:real)% (w:real^3)} INTER ballnorm_fan x)

==> ?s:real. &0 <= s/\ s<= &1
/\ y1=inv(norm((&1 - s) % v + s % ((&1 - t) % u + t % w)-x))%((&1 - s) % v + s % ((&1 - t) % u + t % w)-x)+x`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"A")
THEN MP_TAC(ISPECL[`(x:real^3)` ;` (v:real^3)`;`(&1 - t) % (u:real^3) + t % (w:real^3)`;]th3)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(x:real^3)` ;` (v:real^3)`;`(&1 - t) % (u:real^3) + t % (w:real^3)`;]AFF_GT_1_2)
THEN RESA_TAC
THEN REMOVE_THEN"A" MP_TAC
THEN ASM_REWRITE_TAC[INTER;IN_ELIM_THM;dist;ballnorm_fan]
THEN STRIP_TAC
THEN SUBGOAL_THEN`~(&1 - t1= &0)/\ &0 < &1-t1` ASSUME_TAC
THENL[
REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ISPECL[`t2:real`;`t3:real`]REAL_LT_ADD)
THEN REAL_ARITH_TAC;

MP_TAC(ISPEC`(&1 -t1:real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(&1-t1:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`inv ((&1 - t1:real))`;`(t2:real)`]REAL_LT_MUL)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`inv ((&1 - t1:real))`;`(t3:real)`]REAL_LT_MUL)
THEN RESA_TAC
THEN FIND_ASSUM MP_TAC `t1+t2+t3= &1:real`
THEN REWRITE_TAC[REAL_ARITH`A+B+C=D<=>B+C=D-A`]
THEN DISCH_TAC
THEN MP_TAC(SET_RULE`t2+t3 = &1-t1 ==> (inv (&1-t1))*(t2+t3) = (inv (&1-t1))* (&1-t1:real)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C)=D<=>D- A*C=A*B`]
THEN REDUCE_ARITH_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`inv ((&1 - t1:real))*(t3:real)`
THEN STRIP_TAC
THENL[
ASM_TAC
THEN REAL_ARITH_TAC;
STRIP_TAC
THENL[
ASM_TAC
THEN REAL_ARITH_TAC;
ASM_REWRITE_TAC[VECTOR_ARITH`(inv (&1 - t1) * t2) % v +
  (inv (&1 - t1) * t3) % ((&1 - t) % u + t % w)-x= inv (&1 - t1) %( t2 % v + t3 % ((&1 - t) % u + t % w))-x:real^3`]
THEN FIND_ASSUM MP_TAC `norm(x-y1:real^3)= &1`
THEN FIND_ASSUM MP_TAC `y1 = t1 % x + t2 % v + t3 % ((&1 - t) % u + t % w):real^3`
THEN DISCH_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[NORM_SUB]
THEN REWRITE_TAC[VECTOR_ARITH`(B%A+C+D)-A=C+D-(&1-B)%A`]
THEN DISCH_TAC
THEN MP_TAC(SET_RULE`norm (t2 % v + t3 % ((&1 - t) % u + t % w) - (&1 - t1) % x:real^3) = &1 ==> (inv (&1-t1))*(norm (t2 % v + t3 % ((&1 - t) % u + t % w) - (&1 - t1) % x) ) = (inv (&1-t1))* (&1):real`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN MP_TAC(REAL_ARITH`&0<inv(&1-t1:real)==> &0<= inv(&1-t1:real)`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`inv(&1-t1:real)`;`t2 % v + t3 % ((&1 - t) % u + t % w) - (&1 - t1) % x:real^3`]NORM_MUL)
THEN MP_TAC(ISPEC`inv(&1-t1:real)`REAL_ABS_REFL)
THEN RESP_TAC
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%(B+C-E%D)=A%(B+C)-(A*E)%D`]
THEN REDUCE_VECTOR_TAC
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[(th)])
THEN ASM_REWRITE_TAC[REAL_INV_INV;VECTOR_ARITH`(&1-A)%(B%C-D)+D= A%D+(B*(&1-A))%C:real^3`]
THEN REDUCE_VECTOR_TAC]]]);;


let separate1_sphere_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) (w:real^3) (v1:real^3) (u1:real^3).

FAN(x,V,E) /\  {v,u} INTER {v1,u1}={} /\ {v,u} IN E/\ {u,w} IN E /\ {v1,u1} IN E
/\ ~(coplanar{x,v,u,w})
==>
?h:real.
(&0 < h) /\ (h<= &1)
/\ 
(!t:real. &0<t /\ t<h 
==> aff_gt{x} {v,(&1-t)%u+ t % w} INTER aff_ge {x} {v1, u1} INTER ballnorm_fan x={})`,

REPEAT STRIP_TAC 
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`] 
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] 
THEN MRESA_TAC exist_close_fan[`(x:real^3)` ;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (v:real^3)`;`(u:real^3)`;
`(v1:real^3)`;` (u1:real^3)`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THA")
THEN MRESA_TAC exists_open_not_collinear[`(x:real^3)` ;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (v:real^3)`;`(u:real^3)`;`(w:real^3)`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"A" )
THEN MRESA_TAC exists_point_small_edges_fan [`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(v:real^3)`;`(u:real^3) `;`(w:real^3)`;`h:real`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"B" )
THEN EXISTS_TAC`min ((h':real)/ &2) (t1:real)`
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*1*)[
ASM_TAC
THEN REAL_ARITH_TAC;(*1*)

STRIP_TAC
THENL(*2*)[

ASM_TAC
THEN REAL_ARITH_TAC;(*2*)

REPEAT STRIP_TAC
THEN REWRITE_TAC[EXTENSION]
THEN GEN_TAC
THEN EQ_TAC
THENL(*3*)[

STRIP_TAC
THEN REWRITE_TAC[EMPTY;IN_ELIM_THM]
THEN SUBGOAL_THEN `!y1:real^3.
          y1 IN aff_gt {x} {v, (&1-t)%u+(t:real)% (w:real^3)} INTER ballnorm_fan x
==> ?y2:real^3.        y2 IN aff_ge {x:real^3} {(v:real^3), (u:real^3)} INTER ballnorm_fan x /\
          dist (y1,y2)<(h:real)` ASSUME_TAC
THENL(*4*)[
MP_TAC(REAL_ARITH`&0 <(t:real)/\ &0< h' /\ t < min ((h':real)/ &2) (t1:real)==> &0 <= t /\ t<=t1 /\ t< h'`)
THEN RESA_TAC
THEN REMOVE_THEN"A" (fun th-> MP_TAC(ISPEC`t:real`th))
THEN RESA_TAC
THEN REPEAT STRIP_TAC
THEN MRESA_TAC same_projective_sphere_gt_fan [`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(v:real^3)`;`(u:real^3) `;`(w:real^3)`;` (t:real) `;`(y1:real^3)`]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - t) % (u:real^3) + t % (w:real^3)`;]
THEN MRESA_TAC AFF_GE_1_2[`(x:real^3)` ;` (v:real^3)`;`(u:real^3) `;]
THEN ASM_REWRITE_TAC[INTER;IN_ELIM_THM;dist]
THEN MRESA_TAC expansion1_convex_fan[`(v:real^3)`;` (u:real^3)`;` s:real`]
THEN MRESA_TAC origin_point_not1_in_convex_fan[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;`(u:real^3)`]
THEN SUBGOAL_THEN`~(x=(&1-s)%v+s%u:real^3)` ASSUME_TAC
THENL(*5*)[
POP_ASSUM MP_TAC
THEN MATCH_MP_TAC MONO_NOT
THEN RESA_TAC;(*5*)

MP_TAC(ISPECL[`x:real^3`;`(&1 - s) % v + s % u:real^3`]imp_norm_not_zero_fan)
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[NORM_SUB]
THEN REWRITE_TAC[VECTOR_ARITH`(A+B)-C=A+B-C:real^3`]
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`(&1 - s) % v + s % u-x:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(&0 =norm((&1 - s) % v + s % u-x:real^3)) /\ &0 <= norm((&1 - s) % v + s % u-x:real^3)==> &0 <norm((&1 - s) % v + s % u-x:real^3)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(norm((&1 - s) % v + s % u-x:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(norm((&1 - s) % v + s % u-x:real^3))`REAL_LE_INV)
THEN RESA_TAC
THEN EXISTS_TAC`inv (norm ((&1 - s) % v + s % u - x)) %
      ((&1 - s) % v + s %  u - x) +x:real^3`
THEN STRIP_TAC
THENL(*6*)[
STRIP_TAC
THENL(*7*)[
EXISTS_TAC`&1 - inv (norm ((&1 - s) % v + s % u - x:real^3)) `
THEN EXISTS_TAC`inv (norm ((&1 - s) % v + s % u - x:real^3)) * (&1 - s)`
THEN EXISTS_TAC`inv (norm ((&1 - s) % v + s % u - x:real^3)) * (s)`
THEN REWRITE_TAC[VECTOR_ARITH`A%(B%C + a%b-c)+c=(&1-A)%c+(A*B)%C+(A*a)%b:real^3`]
THEN ASM_REWRITE_TAC[REAL_ARITH`&1-A+A*(&1-C)+A*C= &1`]
THEN STRIP_TAC
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REAL_ARITH_TAC;(*7*)

REWRITE_TAC[ballnorm_fan;IN_ELIM_THM;dist;VECTOR_ARITH`A-(B+A)=(--B):real^3`;NORM_NEG;NORM_MUL]
THEN MP_TAC(ISPEC`inv (norm ((&1 - s) % v + s % u - x:real^3))`REAL_ABS_REFL)
THEN RESP_TAC
THEN ASM_REWRITE_TAC[]](*7*);(*6*)

ASM_REWRITE_TAC[VECTOR_ARITH`inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
  ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x) +
  x -
  (inv (norm ((&1 - s) % v + s % u - x)) % ((&1 - s) % v + s % u - x) + x)=
inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
  ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x) -
  (inv (norm ((&1 - s) % v + s % u - x)) % ((&1 - s) % v + s % u - x) ):real^3`]
THEN REMOVE_THEN "B" (fun th-> MP_TAC(ISPEC`t:real`th))
THEN RESA_TAC
THEN POP_ASSUM (fun th-> MP_TAC(ISPEC`s:real`th))
THEN RESA_TAC
THEN ASM_REWRITE_TAC[NORM_SUB]](*6*)](*5*);(*4*)
POP_ASSUM (fun th->MP_TAC(ISPEC`x':real^3`th))
THEN MP_TAC(SET_RULE` x' IN aff_gt {x} {v, (&1 - t) % u + t % w} INTER aff_ge {x} {v1, u1} INTER
      ballnorm_fan x ==>  x' IN aff_ge {x} {v1, u1:real^3} INTER ballnorm_fan x
/\ x' IN aff_gt {x} {v, (&1 - t) % u + t % w:real^3} INTER ballnorm_fan x`)
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN SIMP_TAC[]
THEN REPEAT STRIP_TAC
THEN REMOVE_THEN "THA" (fun th->  MP_TAC(ISPECL[`y2:real^3`;`x':real^3`]th))
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN SIMP_TAC[dist;NORM_SUB]
THEN REAL_ARITH_TAC](*4*);(*3*)

SET_TAC[]]]]);;









let SCALE_AFF_TAC th=REPEAT GEN_TAC
THEN DISCH_TAC
THEN REPEAT GEN_TAC
THEN MRESAL_TAC  th[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`&1 - a* t2- a* t3:real`
THEN EXISTS_TAC`a* t2:real`
THEN EXISTS_TAC`a* t3:real`
THEN ONCE_REWRITE_TAC[TAUT`A/\B/\C/\D<=>D/\ A/\B/\C`]
THEN STRIP_TAC
THENL[
ASM_REWRITE_TAC[]
THEN FIND_ASSUM MP_TAC `t1+t2+t3= &1:real`
THEN REWRITE_TAC[REAL_ARITH`A+B+C= &1<=>A= &1- B -C:real`]
THEN DISCH_TAC
THEN ASM_REWRITE_TAC[]
THEN VECTOR_ARITH_TAC;

MP_TAC(ISPECL[`a:real`;`(t2:real)`]REAL_LT_MUL)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`a:real`;`(t3:real)`]
THEN MRESA_TAC REAL_LE_MUL[`a:real`;`(t2:real)`]
THEN MRESA_TAC REAL_LE_MUL[`a:real`;`(t3:real)`]
THEN ASM_TAC
THEN REAL_ARITH_TAC];;


let scale_aff_ge_fan=prove(`!x:real^3 v:real^3 u:real^3.
     DISJOINT {x} {v,u}
==>
(!y:real^3 a:real. y IN  aff_ge {x} {v,u} /\ &0 <= a==> a%(y-x)+x IN aff_ge{x} {v,u})`,
SCALE_AFF_TAC AFF_GE_1_2);;
let scale_aff_gt_fan=prove(`!x:real^3 v:real^3 u:real^3.
     DISJOINT {x} {v,u}
==>
(!y:real^3 a:real. y IN  aff_gt {x} {v,u} /\ &0 < a==> a%(y-x)+x IN aff_gt{x} {v,u})`,
SCALE_AFF_TAC AFF_GT_1_2);;


let origin_is_not_aff_gt_fan=prove(`!x:real^3 v:real^3 u:real^3.
   ~(u IN aff {x,v}) /\ DISJOINT {x} {v,u}==> ~(x IN  aff_gt {x} {v,u})`,
REPEAT GEN_TAC
THEN STRIP_TAC
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN REWRITE_TAC[GSYM FORALL_NOT_THM;DE_MORGAN_THM]
THEN REPEAT STRIP_TAC 
THEN DISJ_CASES_TAC(REAL_ARITH`~( &0< (t2:real))\/ ( &0< (t2:real))`)
THENL[ASM_MESON_TAC[];
DISJ_CASES_TAC(REAL_ARITH`~( &0< (t3:real))\/ ( &0< (t3:real))`)
THENL[ASM_MESON_TAC[];
DISJ_CASES_TAC(REAL_ARITH`~( t1+t2+(t3:real)= &1)\/ (  t1+t2+(t3:real)= &1)`)
THENL[ASM_MESON_TAC[];
ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`A+B+C= &1<=>A= &1- B -C:real`]
THEN DISCH_TAC
THEN MP_TAC(REAL_ARITH`&0<t3:real==> ~(t3= &0)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(t3:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REWRITE_TAC[VECTOR_ARITH`A=( &1-B-C) %A+B%E+C%D <=> C%(D-A)= (--B)%(E-A)`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`t3 % (u - x) = (--t2) % (v - x):real^3 ==> (inv (t3))%(t3 % (u - x)) = (inv (t3))%((--t2) % (v - x))`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[VECTOR_ARITH`A-B=C%(V-B)<=>A=( &1-C)%B+C%V`]
THEN FIND_ASSUM MP_TAC `~(u IN aff {x,v}:real^3->bool)`
THEN MATCH_MP_TAC MONO_NOT
THEN STRIP_TAC
THEN REWRITE_TAC[aff; AFFINE_HULL_2; IN_ELIM_THM]
THEN EXISTS_TAC`&1 - inv t3 * --t2:real`
THEN EXISTS_TAC`inv t3 * --t2:real`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC]]]);;
let fan_run_in_small1_is_fan=prove(
`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 v1:real^3 u1:real^3.
FAN(x,V,E) /\  {v,u} INTER {v1,u1}={} /\ {v,u} IN E/\ {u,w} IN E /\ {v1,u1} IN E
/\ ~(coplanar{x,v,u,w})
==> 
?t1:real. &0 < t1 /\ t1 <= &1
/\ (!t:real. &0< t /\ t< t1 ==> aff_gt{x} {v,(&1-t)%u+ t % w} INTER aff_ge{x} {v1,u1}={})`,

REPEAT STRIP_TAC 
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u1:real^3)`;
` (v1:real^3)`]  
THEN MRESA_TAC exists_open_not_collinear[`(x:real^3)` ;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (v:real^3)`;`(u:real^3)`;`(w:real^3)`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"BA")
THEN MRESA_TAC separate1_sphere_fan[`(x:real^3)`;` (V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (v:real^3) `;`(u:real^3)`;` (w:real^3)`;` (v1:real^3)`;`(u1:real^3)`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"A")
THEN REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH` &0 < t1/\ &0 < h ==> &0 < min (h:real) (t1:real)`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`  t1 <= &1  /\ h <= &1==>  min (h:real) (t1:real)<= &1`)
THEN RESA_TAC
THEN EXISTS_TAC`min (h:real) (t1:real)`
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`  &0 < t /\ t<  min (h:real) (t1:real)==> t< h /\ t<= t1 /\ &0 <= t`)
THEN RESA_TAC
THEN REMOVE_THEN "A"(fun th-> MRESA1_TAC th `(t:real)`)
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"A")
THEN REMOVE_THEN "BA"(fun th-> MRESA1_TAC th `(t:real)`)
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - t) % (u:real^3) + t % (w:real^3)`;]
THEN MRESA_TAC origin_is_not_aff_gt_fan[`x:real^3`;`v:real^3`;`(&1-t)%u+t%w:real^3`]
THEN REWRITE_TAC[EXTENSION;INTER;IN_ELIM_THM]
THEN GEN_TAC
THEN EQ_TAC
THENL[ REPEAT STRIP_TAC
THEN SUBGOAL_THEN`~(x=x':real^3)`ASSUME_TAC
THENL[
POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];
MRESA_TAC imp_norm_not_zero_fan[`x':real^3`;`x:real^3`]
THEN ASSUME_TAC(ISPEC`x'-x:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(norm(x'-x:real^3)= &0) /\ &0 <= norm(x'-x:real^3)==> &0 <norm(x'-x:real^3)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LE_INV `norm(x'-x:real^3)`
THEN MRESA1_TAC REAL_LT_INV `norm(x'-x:real^3)`
THEN MRESA1_TAC REAL_MUL_LINV `norm(x'-x:real^3)`
THEN MRESA_TAC scale_aff_ge_fan[`x:real^3`;`v1:real^3`;`u1:real^3`]
THEN POP_ASSUM(fun th -> MRESA_TAC th [`x':real^3`;`inv(norm(x'-x:real^3))`])
THEN MRESA_TAC scale_aff_gt_fan[`x:real^3`;`v:real^3`;`(&1-t)%u+t%w:real^3`]
THEN POP_ASSUM(fun th -> MRESA_TAC th [`x':real^3`;`inv(norm(x'-x:real^3))`])
THEN SUBGOAL_THEN`inv (norm (x' - x)) % (x' - x) + x IN ballnorm_fan x:real^3->bool` ASSUME_TAC
THENL[
MRESA1_TAC REAL_ABS_REFL `inv(norm(x'-x:real^3))`
THEN ASM_REWRITE_TAC[ballnorm_fan;IN_ELIM_THM;dist;VECTOR_ARITH`B-(A+B)= --A:real^3`;NORM_NEG;NORM_MUL];

ASM_TAC
THEN SET_TAC[]]];

SET_TAC[]]);;







let azim_line_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 t:real^1.
~coplanar {x,v,u,(&1-drop(t))%u+drop(t)%w}
==> (\(t:real^1). azim x v u ((&1 - drop(t)) % u + drop(t) % w)) real_continuous at t`,
REPEAT STRIP_TAC
THEN REWRITE_TAC[REAL_CONTINUOUS_CONTINUOUS1; o_DEF; CONTINUOUS_AT_LIFT_COMPONENT]
THEN MP_TAC(ISPECL[`(\(t:real^1). ((&1 - drop(t)) % (u:real^3) + drop(t) %(w:real^3)))`;`(\(w:real^3). lift(azim (x:real^3) (v:real^3) (u:real^3) w))`;`t:real^1`] CONTINUOUS_AT_COMPOSE)
THEN REWRITE_TAC[o_DEF]
THEN DISCH_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN REWRITE_TAC[ISPECL[`u:real^3`;`w:real^3`;`t:real^1`]collinear1_continuous_fan]
THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`;`((&1 - drop(t)) % (u:real^3) + drop(t) %(w:real^3))`]REAL_CONTINUOUS_AT_AZIM)
THEN REWRITE_TAC[REAL_CONTINUOUS_CONTINUOUS1; o_DEF; CONTINUOUS_AT_LIFT_COMPONENT]
THEN ASM_MESON_TAC[]);;



(******fully surrounded*******)

let fan81=new_definition`fan81 (x:real^3,(V:real^3->bool),(E:(real^3->bool)->bool)):bool<=>(!v:real^3 u:real^3. {v,u} IN E ==> azim_fan x V E v u <pi)`;;
let fan80=new_definition`fan80 (x:real^3,(V:real^3->bool),(E:(real^3->bool)->bool)):bool<=>(!v:real^3 u:real^3. {v,u} IN E ==> &0< azim x v u (sigma_fan x V E v u) /\ azim x v u (sigma_fan x V E v u) <pi)`;;









let continuous_coplanar_fan=prove(`!x:real^3  v:real^3 u:real^3 w:real^3.
 ~(coplanar{x,v,u,w})
==>(!t:real.  ~(t= &0) ==> ~coplanar {x,v,u,(&1-t)%u+t%w} )`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[]
THEN FIND_ASSUM MP_TAC`~(coplanar{x,v,u,w:real^3})`
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[COPLANAR_DET_EQ_0;VECTOR_ARITH`((&1 - t) % u + t % w) - x=(&1 - t) % (u-x) + t % (w - x):real^3`;DET_3;VECTOR_3;VECTOR_ADD_COMPONENT;VECTOR_MUL_COMPONENT
;REAL_ARITH`(v - x)$1 * (u - x)$2 * ((&1 - t) * (u - x)$3 + t * (w - x)$3) +
 (v - x)$2 * (u - x)$3 * ((&1 - t) * (u - x)$1 + t * (w - x)$1) +
 (v - x)$3 * (u - x)$1 * ((&1 - t) * (u - x)$2 + t * (w - x)$2) -
 (v - x)$1 * (u - x)$3 * ((&1 - t) * (u - x)$2 + t * (w - x)$2) -
 (v - x)$2 * (u - x)$1 * ((&1 - t) * (u - x)$3 + t * (w - x)$3) -
 (v - x)$3 * (u - x)$2 * ((&1 - t) * (u - x)$1 + t * (w - x)$1)=
t*((v - x)$1 * (u - x)$2 * ((w - x)$3) +
 (v - x)$2 * (u - x)$3 * ( (w - x)$1) +
 (v - x)$3 * (u - x)$1 * ((w - x)$2) -
 (v - x)$1 * (u - x)$3 * ((w - x)$2) -
 (v - x)$2 * (u - x)$1 * ( (w - x)$3) -
 (v - x)$3 * (u - x)$2 * ( (w - x)$1)):real`;REAL_ENTIRE]
THEN ASM_TAC
THEN REAL_ARITH_TAC);;




let open_is_not_zero_fan=prove(`open{y:real^1 | ?x. ~(x = &0) /\ y = lift x}`,
(let equality_real_fan=prove(`{y:real^1 | ?x. ~(x = &0) /\ y = lift x}={y:real^1 | ~(drop y = &0)}`,
REWRITE_TAC[EXTENSION;IN_ELIM_THM]
THEN GEN_TAC
THEN EQ_TAC
THENL[
STRIP_TAC
THEN ASM_REWRITE_TAC[LIFT_DROP];
STRIP_TAC
THEN EXISTS_TAC`drop (x:real^1)`
THEN ASM_REWRITE_TAC[LIFT_DROP]])in
(let ngu=prove(`{x | x IN (:real^1) /\ x = vec 0}={x | x IN (:real^1) /\ x$1 = &0}`,
REWRITE_TAC[EXTENSION;IN_ELIM_THM]
THEN GEN_TAC
THEN EQ_TAC
THENL[STRIP_TAC
THEN ASM_REWRITE_TAC[VEC_COMPONENT];
SIMP_TAC[IN_UNIV;CART_EQ;LAMBDA_BETA;VEC_COMPONENT;DIMINDEX_1;ARITH_RULE`1<=i /\ i<=1<=>i=1`]])
in
REWRITE_TAC[equality_real_fan]
THEN REWRITE_TAC[OPEN_CLOSED;DIFF; IN_ELIM_THM;]
THEN MP_TAC(ISPECL[`(\(t:real^1). t)`;`(:real^1)`;`(vec 0):real^1`]CONTINUOUS_CLOSED_PREIMAGE_CONSTANT)
THEN MP_TAC(ISPECL[`0`;`1`]VEC_COMPONENT)
THEN SIMP_TAC[CONTINUOUS_ON_ID;CLOSED_UNIV; DIMINDEX_1;drop;ngu])));;










let azim_continuous_when_not_coplanar=prove(`!x:real^3 v:real^3 u:real^3 w:real^3.
~coplanar {x,v,u,w}
==> (\(t:real). azim x v u ((&1 - t) % u + t % w)) real_continuous_on {t:real| ~(t= &0)}`,
REWRITE_TAC[REAL_CONTINUOUS_ON;o_DEF;IMAGE;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASSUME_TAC(open_is_not_zero_fan)
THEN MRESA_TAC CONTINUOUS_ON_EQ_CONTINUOUS_AT[`(\t:real^1. lift (azim x v u ((&1 - drop t) % u + drop t % w))):real^1->real^1`;`{y:real^1| ?x. ~(x = &0) /\ y = lift x}`]
THEN REWRITE_TAC[IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[LIFT_DROP]
THEN MRESAL_TAC azim_line_fan[`x:real^3`;` v:real^3`;` u:real^3`;` w:real^3`;` (lift x''):real^1`][REAL_CONTINUOUS_CONTINUOUS1; o_DEF;LIFT_DROP]
THEN POP_ASSUM MATCH_MP_TAC
THEN MRESA_TAC continuous_coplanar_fan[`x:real^3 `;` v:real^3`;` u:real^3`;` w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `x'':real`));;

let injective_azim_coplanar=prove(`!x:real^3 v:real^3 u:real^3 w:real^3.
~coplanar {x,v,u,w}
==> 
!a:real b:real. ~(a= &0) /\ ~(b= &0)/\ (\(t:real). azim x v u ((&1 - t) % u + t % w))a=(\(t:real). azim x v u ((&1 - t) % u + t % w))b==>a=b`,
REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MRESA_TAC continuous_coplanar_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM(fun th -> MP_TAC(ISPEC`a:real`th)THEN ASSUME_TAC(th))
THEN POP_ASSUM(fun th -> MP_TAC(ISPEC`b:real`th))
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,B,D,C}`]
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MRESA_TAC NOT_COPLANAR_NOT_COLLINEAR[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC NOT_COPLANAR_NOT_COLLINEAR[`x:real^3`;`v:real^3`;`(&1 - a) % u + a % w:real^3`;`u:real^3`]
THEN MRESA_TAC NOT_COPLANAR_NOT_COLLINEAR[`x:real^3`;`v:real^3`;`(&1 - b) % u + b % w:real^3`;`u:real^3`]
THEN MRESA_TAC AZIM_EQ [`x:real^3`;`v:real^3`;`u:real^3`;`(&1 - b) % u + b % w:real^3`;`(&1 - a) % u + a % w:real^3`;]
THEN MRESA_TAC AZIM_EQ_0 [`x:real^3`;`v:real^3`;`(&1 - a) % u + a % w:real^3`;`(&1 - b) % u + b % w:real^3`;]
THEN MRESA_TAC AZIM_EQ_0_PI_IMP_COPLANAR [`x:real^3`;`v:real^3`;`(&1 - a) % u + a % w:real^3`;`(&1 - b) % u + b % w:real^3`;]
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`~coplanar {x,v,u,w:real^3}`
THEN REWRITE_TAC[COPLANAR_DET_EQ_0;VECTOR_ARITH`((&1 - t) % u + t % w) - x=(&1 - t) % (u-x) + t % (w - x):real^3`;DET_3;VECTOR_3;VECTOR_ADD_COMPONENT;VECTOR_MUL_COMPONENT]
THEN STRIP_TAC
THEN REWRITE_TAC[REAL_ARITH`(v - x)$1 *
 ((&1 - a) * (u - x)$2 + a * (w - x)$2) *
 ((&1 - b) * (u - x)$3 + b * (w - x)$3) +
 (v - x)$2 *
 ((&1 - a) * (u - x)$3 + a * (w - x)$3) *
 ((&1 - b) * (u - x)$1 + b * (w - x)$1) +
 (v - x)$3 *
 ((&1 - a) * (u - x)$1 + a * (w - x)$1) *
 ((&1 - b) * (u - x)$2 + b * (w - x)$2) -
 (v - x)$1 *
 ((&1 - a) * (u - x)$3 + a * (w - x)$3) *
 ((&1 - b) * (u - x)$2 + b * (w - x)$2) -
 (v - x)$2 *
 ((&1 - a) * (u - x)$1 + a * (w - x)$1) *
 ((&1 - b) * (u - x)$3 + b * (w - x)$3) -
 (v - x)$3 *
 ((&1 - a) * (u - x)$2 + a * (w - x)$2) *
 ((&1 - b) * (u - x)$1 + b * (w - x)$1)=
(b-a)*((v - x)$1 * (u - x)$2 * (w - x)$3 +
   (v - x)$2 * (u - x)$3 * (w - x)$1 +
   (v - x)$3 * (u - x)$1 * (w - x)$2 -
   (v - x)$1 * (u - x)$3 * (w - x)$2 -
   (v - x)$2 * (u - x)$1 * (w - x)$3 -
   (v - x)$3 * (u - x)$2 * (w - x)$1)`;REAL_ENTIRE]
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC);;











let fan_run_in_small21_is_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ {v,w1} IN E 
/\ ~coplanar {x,v,u,w}
==> 
?t1:real. &0<t1 /\ t1<= &1
/\ (!t:real. &0< t /\ t<= t1==> aff_gt{x} {v,(&1-t)%u+ t % w} INTER aff_ge{x} {v,w1}={})`,

REPEAT STRIP_TAC
THEN MRESA_TAC exists_open_not_collinear[`(x:real^3)` ;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (v:real^3)`;`(u:real^3)`;`(w:real^3)`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"A")
THEN DISJ_CASES_TAC(SET_RULE`(?h:real. &0<h /\ h<= &1/\ azim x v u w1=azim x v u ((&1-h)%u+h%w:real^3))\/ ~(?h:real. &0<h /\ h<= &1 /\ azim x v u ((&1-h)%u+h%w:real^3)=azim x v u w1)`)
THENL(*1*)[
POP_ASSUM MP_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`min (t1:real) ((h:real)/ &2)`
THEN STRIP_TAC
THENL(*2*)[
ASM_TAC
THEN REAL_ARITH_TAC;(*2*)

STRIP_TAC
THENL(*3*)[ASM_TAC
THEN REAL_ARITH_TAC;(*3*)
REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0<t:real==> &0<=t`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`(t:real)<= min (t1:real) (h:real/ &2) ==> t <= t1`)
THEN RESA_TAC
THEN REMOVE_THEN"A"(fun th-> MRESA1_TAC th `t:real`)
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - t) % (u:real^3) + t % (w:real^3)`;]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w1:real^3)`;
` (v:real^3)`]  
THEN REWRITE_TAC[INTER;EXTENSION;IN_ELIM_THM;]
THEN GEN_TAC
THEN EQ_TAC
THENL(*4*)[
DISCH_TAC
THEN SUBGOAL_THEN `(&1-t)%u+t%w IN aff_ge {x,v} {w1}:real^3->bool` ASSUME_TAC
THENL(*5*)[
POP_ASSUM MP_TAC
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v:real^3`;`w1:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GE_2_1[`x:real^3`;`v:real^3`;`w1:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`(&1-t)%u +t%w:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2 % v + t3 % ((&1 - t) % u + t % w) =
 t1'' % x + t2' % v + t3' % w1<=> t3 % ((&1 - t) % u + t % w) =
 (t1''-t1') % x + (t2'-t2) % v + t3' % w1:real^3`]
THEN MP_TAC(REAL_ARITH`&0<t3:real==> ~(t3= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`(t3:real)`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t3 % ((&1 - t) % u + t % w) = (t1''-t1') % x + (t2'-t2) % v + t3' % w1:real^3 
==> (inv (t3))%(t3 % ((&1 - t) % u + t % w)) = (inv (t3))%((t1''-t1') % x + (t2'-t2) % v + t3' % w1)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`;VECTOR_ARITH`A%(B%X+C%Y+D%Z)=(A*B)%X+(A*C)%Y+(A*D)%Z`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC`t1'' + t2' + t3' = &1:real`
THEN FIND_ASSUM MP_TAC`t1' + t2 + t3 = &1:real`
THEN DISCH_TAC 
THEN POP_ASSUM (fun th -> GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SYM(th)])
THEN REWRITE_TAC[REAL_ARITH`t1'' + t2' + t3' = t1' + t2 + t3<=> (t1''-t1') + (t2'-t2) + t3' = t3:real`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
(t1''-t1') + (t2'-t2) + t3' = t3
==> (inv (t3))*((t1''-t1') + (t2'-t2) + t3') = (inv (t3))*t3`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C+D)=A*B+A*C+A*D`]
THEN STRIP_TAC
THEN EXISTS_TAC`inv t3 * (t1'' - t1'):real`
THEN EXISTS_TAC`inv t3 * (t2' - t2):real`
THEN EXISTS_TAC`inv t3 * t3':real`
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LE_INV
THEN ASM_TAC
THEN REAL_ARITH_TAC;(*5*)
MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`v:real^3`;`(&1 - t) % u + t % w:real^3`;`w1:real^3`;]
THEN MRESA_TAC AZIM_EQ_0_ALT[`x:real^3`;`v:real^3`;`(&1 - t) % u + t % w:real^3`;`w1:real^3`;]
THEN MRESA_TAC AZIM_EQ_ALT[`x:real^3`;`v:real^3`;`u:real^3`;`w1:real^3`;`(&1 - t) % u + t % w:real^3`;]
THEN MRESA_TAC injective_azim_coplanar[`x:real^3`;` v:real^3`;` u:real^3`;` w:real^3`]
THEN POP_ASSUM (fun th->MRESA_TAC th [`h:real`;`t:real`])
THEN ASM_TAC
THEN REAL_ARITH_TAC](*5*)(*4*);
SET_TAC[]]]];(*1*)
POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[NOT_EXISTS_THM]
THEN DISCH_THEN(LABEL_TAC"B")
THEN EXISTS_TAC`t1:real`
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0<t:real==> &0<=t`)
THEN RESA_TAC
THEN REMOVE_THEN"A"(fun th-> MRESA1_TAC th `t:real`)
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - t) % (u:real^3) + t % (w:real^3)`;]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w1:real^3)`;
` (v:real^3)`]  
THEN REWRITE_TAC[INTER;EXTENSION;IN_ELIM_THM;]
THEN GEN_TAC
THEN EQ_TAC
THENL(*2*)[
DISCH_TAC
THEN SUBGOAL_THEN `(&1-t)%u+t%w IN aff_ge {x,v} {w1}:real^3->bool` ASSUME_TAC
THENL(*3*)[
POP_ASSUM MP_TAC
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v:real^3`;`w1:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GE_2_1[`x:real^3`;`v:real^3`;`w1:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`(&1-t)%u +t%w:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2 % v + t3 % ((&1 - t) % u + t % w) =
 t1'' % x + t2' % v + t3' % w1<=> t3 % ((&1 - t) % u + t % w) =
 (t1''-t1') % x + (t2'-t2) % v + t3' % w1:real^3`]
THEN MP_TAC(REAL_ARITH`&0<t3:real==> ~(t3= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`(t3:real)`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t3 % ((&1 - t) % u + t % w) = (t1''-t1') % x + (t2'-t2) % v + t3' % w1:real^3 
==> (inv (t3))%(t3 % ((&1 - t) % u + t % w)) = (inv (t3))%((t1''-t1') % x + (t2'-t2) % v + t3' % w1)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`;VECTOR_ARITH`A%(B%X+C%Y+D%Z)=(A*B)%X+(A*C)%Y+(A*D)%Z`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC`t1'' + t2' + t3' = &1:real`
THEN FIND_ASSUM MP_TAC`t1' + t2 + t3 = &1:real`
THEN DISCH_TAC 
THEN POP_ASSUM (fun th -> GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SYM(th)])
THEN REWRITE_TAC[REAL_ARITH`t1'' + t2' + t3' = t1' + t2 + t3<=> (t1''-t1') + (t2'-t2) + t3' = t3:real`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
(t1''-t1') + (t2'-t2) + t3' = t3
==> (inv (t3))*((t1''-t1') + (t2'-t2) + t3') = (inv (t3))*t3`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C+D)=A*B+A*C+A*D`]
THEN STRIP_TAC
THEN EXISTS_TAC`inv t3 * (t1'' - t1'):real`
THEN EXISTS_TAC`inv t3 * (t2' - t2):real`
THEN EXISTS_TAC`inv t3 * t3':real`
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LE_INV
THEN ASM_TAC
THEN REAL_ARITH_TAC;(*3*)
MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`v:real^3`;`(&1 - t) % u + t % w:real^3`;`w1:real^3`;]
THEN MRESA_TAC AZIM_EQ_0_ALT[`x:real^3`;`v:real^3`;`(&1 - t) % u + t % w:real^3`;`w1:real^3`;]
THEN MRESA_TAC AZIM_EQ_ALT[`x:real^3`;`v:real^3`;`u:real^3`;`w1:real^3`;`(&1 - t) % u + t % w:real^3`;]
THEN REMOVE_THEN "B"(fun th -> MRESAL1_TAC th `t:real`[DE_MORGAN_THM])
THEN ASM_TAC
THEN REAL_ARITH_TAC];
SET_TAC[]]]);;


let fan_run_in_small2_is_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 w1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ {v,w1} IN E 
/\ ~coplanar {x,v,u,w}
==> 
?t1:real. &0<t1 /\ t1<= &1
/\ (!t:real. &0< t /\ t< t1==> aff_gt{x} {v,(&1-t)%u+ t % w} INTER aff_ge{x} {v,w1}={})`,
REPEAT STRIP_TAC
THEN MRESA_TAC fan_run_in_small21_is_fan [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;` u:real^3`;` w:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"A")
THEN EXISTS_TAC`t1:real`
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`t<t1==>t<=t1`)
THEN ASM_REWRITE_TAC[]
THEN REMOVE_THEN"A"(fun th-> MRESA1_TAC th `t:real`));;


let AFF_GT_2_2=prove(`!x u v w.
         DISJOINT {x,  u} {v, w}
         ==> aff_gt {x, u} {v, w} =
             {y | ?t1 t2 t3 t4.
                      &0 < t3 /\
                      &0 < t4 /\
                      t1 + t2 + t3 +t4= &1 /\
                      y = t1 % x + t2 %u + t3 % v + t4 % w}`,
AFF_TAC);;


let extension_in_aff_2_2_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
==> 
(!t:real. &0< t /\ t< &1
==> (!t1:real t2:real t3:real. &0<t3  /\ &0<t2 /\ t1+t2+t3= &1
==>t1 % x + t2 % v + t3 % ((&1 - t) % u + t % w) IN aff_gt {x,u} {w,v}))`,
REPEAT STRIP_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`] 
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] 
THEN SUBGOAL_THEN `DISJOINT {x,u:real^3} {w,v:real^3}` ASSUME_TAC
THENL[
REWRITE_TAC[DISJOINT_SYM;SET_RULE`{v:real^3,w:real^3}= {v} UNION {w}`;DISJOINT_UNION]
THEN REWRITE_TAC[SET_RULE`{v} UNION {w}={v:real^3,w:real^3}`]
THEN ONCE_REWRITE_TAC[DISJOINT_SYM]
THEN ASM_REWRITE_TAC[];
MRESAL_TAC AFF_GT_2_2[`x:real^3`;`u:real^3`;`w:real^3`;`v:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t3*(&1-t):real`
THEN EXISTS_TAC`t3*(t):real`
THEN EXISTS_TAC`t2:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`t1 +  t3 * (&1 - t) + t3 * t +t2 = t1+t2+t3:real`;VECTOR_ARITH`t1 % x + t2 % v + t3 % ((&1 - t) % u + t % w) =
 t1 % x + (t3 * (&1 - t)) % u  + (t3 * t) % w+ t2 % v:real^3`]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_TAC
THEN REAL_ARITH_TAC]);;


let inequality3_aim_in_convex_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ 
(&0< azim x u w v ) /\ (azim x u w v <pi)
==> 
(!t:real. &0< t /\ t< &1
==> (!t1:real t2:real t3:real. &0<t3  /\ &0<t2 /\ t1+t2+t3= &1
==> &0< azim x u w (t1 % x + t2 % v + t3 % ((&1 - t) % u + t % w))
/\ azim x u w (t1 % x + t2 % v + t3 % ((&1 - t) % u + t % w))< azim x u w v))`,

REPEAT STRIP_TAC
THEN FIND_ASSUM MP_TAC`{v,u} IN (E:(real^3->bool)->bool)`
THEN ONCE_REWRITE_TAC[SET_RULE`{v,u}={u,v}`]
THEN STRIP_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (v:real^3)`;
` (u:real^3)`] 
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] 
THEN MRESA_TAC WEDGE_LUNE_GT[`x:real^3`;` u:real^3`;`w:real^3`;`v:real^3`]
THEN POP_ASSUM (fun th-> MP_TAC(SYM(th)))
THEN DISCH_TAC
THEN MRESA_TAC extension_in_aff_2_2_fan[`x:real^3`;` v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th `t:real`)
THEN POP_ASSUM(fun th-> MRESAL_TAC th [`t1:real`;`t2:real`;`t3:real`][wedge;IN_ELIM_THM]));;


let fan_run_in_small3_is_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 w1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ {u,w1} IN E 
/\ ~coplanar {x,v,u,w} /\ sigma_fan x V E u w=v
/\ (&0< azim x u w v ) /\ (azim x u w v <pi)
==> 
?t1:real. &0<t1 /\ t1<= &1
/\ (!t:real. &0< t /\ t< t1==> aff_gt{x} {v,(&1-t)%u+ t % w} INTER aff_ge{x} {u,w1}={})`,
REPEAT STRIP_TAC
THEN MRESA_TAC exists_open_not_collinear[`(x:real^3)` ;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (v:real^3)`;`(u:real^3)`;`(w:real^3)`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"A")
THEN EXISTS_TAC`t1:real`
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0<t:real==> &0<=t /\ ~(t= &0)`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`t<t1 /\ t1<= &1==>t<=t1 /\ t< &1`)
THEN RESA_TAC
THEN REMOVE_THEN"A"(fun th-> MRESA1_TAC th `t:real`)
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - t) % (u:real^3) + t % (w:real^3)`;]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w1:real^3)`;
` (u:real^3)`]  
THEN REWRITE_TAC[EXTENSION;INTER;IN_ELIM_THM]
THEN GEN_TAC
THEN EQ_TAC
THENL(*1*)[
MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`(&1-t)%u +t%w:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN SUBGOAL_THEN `x' IN (aff_ge {x,u} {w1}:real^3->bool)`ASSUME_TAC
THENL(*2*)[
POP_ASSUM MP_TAC
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`u:real^3`;`w1:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GE_2_1[`x:real^3`;`u:real^3`;`w1:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`t1'':real`
THEN EXISTS_TAC`t2':real`
THEN EXISTS_TAC`t3':real`
THEN ASM_REWRITE_TAC[];(*2*)

POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MRESA_TAC inequality3_aim_in_convex_fan
[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;` w:real^3`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th `t:real`)
THEN POP_ASSUM(fun th-> MRESA_TAC th [`t1':real`;`t2:real`;`t3:real`])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"BE")
THEN DISCH_THEN(LABEL_TAC"YEU")
THEN SUBGOAL_THEN`~collinear {x, u, x':real^3}`ASSUME_TAC
THENL(*3*)[
MRESA_TAC continuous_coplanar_fan [`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `t:real`)
THEN ASM_REWRITE_TAC[collinear_fan;]
THEN POP_ASSUM MP_TAC
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[coplanar;aff; AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`x:real^3`
THEN EXISTS_TAC`v:real^3`
THEN EXISTS_TAC`u:real^3`
THEN SUBGOAL_THEN`(x:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL(*4*)[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&0`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;(*4*)

SUBGOAL_THEN`(v:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL(*5*)[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC`&0`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;(*5*)

SUBGOAL_THEN`(u:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL(*6*)[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;(*6*)

SUBGOAL_THEN`((&1 - t) % u + t % w:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL(*7*)[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(REAL_ARITH`&0<t3:real==> ~(t3= &0)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(t3:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REWRITE_TAC[VECTOR_ARITH`t1' % x + t2 % v + t3 % ((&1 - t) % u + t % w) = u' % x + v' % u
<=>t3 % ((&1 - t) % u + t % w) = (u'-t1') % x -t2 % v+v' % u:real^3`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t3 % ((&1 - t) % u + t % w) = (u'-t1') % x -t2 % v+v' % u
==> (inv (t3))%(t3 % ((&1 - t) % u + t % w) ) = (inv (t3))%((u'-t1') % x -t2 % v+v' % u):real^3`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%(B-C+D)=A%B-A%C+A%D`;VECTOR_ARITH`A%B%C=(A*B)%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`(inv t3 * (u' - t1')):real`
THEN EXISTS_TAC`-- (inv t3 * t2):real`
THEN EXISTS_TAC`(inv t3 * v'):real`
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A+(--b)%C+d=A-b%C+d`;REAL_ARITH`inv t3 * (u' - t1') + --(inv t3 * t2) + inv t3 * v'=inv t3 * (t3+(u'+v') - (t1'+ t2+t3))`;REAL_ARITH`A+ &1- &1=A`];(*7*)

ASM_TAC
THEN SET_TAC[]](*7*)](*6*)](*5*)](*4*);(*3*)
 
MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`]
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`u:real^3`;`x':real^3`;`w1:real^3`;]
THEN MRESA_TAC AZIM_EQ_0_ALT[`x:real^3`;`u:real^3`;`x':real^3`;`w1:real^3`;]
THEN MRESA_TAC AZIM_EQ_ALT[`x:real^3`;`u:real^3`;`w:real^3`;`w1:real^3`;`x':real^3`;]
THEN REMOVE_THEN"BE" MP_TAC
THEN REMOVE_THEN"YEU" MP_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN FIND_ASSUM MP_TAC`(sigma_fan x V E u w = (v:real^3))`
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN DISJ_CASES_TAC(SET_RULE`(set_of_edge u V E = {w:real^3}) \/ ~(set_of_edge u V E = {w})`)
THENL(*4*)[
FIND_ASSUM MP_TAC`w1 IN set_of_edge (u:real^3) V E`
THEN POP_ASSUM (fun th-> REWRITE_TAC[(th);IN_SING])
THEN DISCH_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC (ISPECL[`x:real^3`;`u:real^3`;`w:real^3`]AZIM_REFL)
THEN REAL_ARITH_TAC;(*4*)

DISJ_CASES_TAC(SET_RULE`(w1:real^3)=w \/ ~((w1:real^3)=w)`)
THENL(*5*)[
ASM_REWRITE_TAC[]
THEN MP_TAC (ISPECL[`x:real^3`;`u:real^3`;`w:real^3`]AZIM_REFL)
THEN REAL_ARITH_TAC;(*5*)

MRESA_TAC SIGMA_FAN[`(x:real^3)`;` (V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(u:real^3)`;`(w:real^3)`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `w1:real^3`)
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC]]]];

SET_TAC[]]);;




let properties_fully_surrounded=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E
/\ (&0< azim x u w v ) /\ (azim x u w v <pi)
==> ~coplanar {x,v,u,w}`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[SET_RULE`{x,v,u,w}={x,u,w,v}`]
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC`{v,u} IN (E:(real^3->bool)->bool)`
THEN REWRITE_TAC[SET_RULE`{v,u}={u,v}`]
THEN STRIP_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (v:real^3)`;
` (u:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`]  
THEN MRESA_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`u:real^3`;`w:real^3`;`v:real^3`]
THEN ASM_TAC
THEN REAL_ARITH_TAC);;






let fan_run_in_small_is_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 v1:real^3 w1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ {v1,w1} IN E 
/\ (&0< azim x u w v ) /\ (azim x u w v <pi)
/\ sigma_fan x V E u w = v
==> 
?t1:real. &0<t1 /\ t1<= &1
/\ (!t:real. &0< t /\ t< t1==> aff_gt{x} {v,(&1-t)%u+ t % w} INTER aff_ge{x} {v1,w1}={})`,
(let lem=prove(`!x v v1. aff_ge {x} {v1, v}=aff_ge {x} {v, v1}`,
REPEAT STRIP_TAC
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`{v,u}={u,v}`]
THEN SIMP_TAC[]) in

REPEAT STRIP_TAC
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN DISJ_CASES_TAC(SET_RULE`(v1=v)\/ ~(v1=v:real^3)`)
THENL[
MRESA_TAC fan_run_in_small2_is_fan [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v1:real^3`;`u:real^3`;` w:real^3`;`w1:real^3`];

DISJ_CASES_TAC(SET_RULE`(v1=u)\/ ~(v1=u:real^3)`)
THENL[
MRESA_TAC fan_run_in_small3_is_fan[`x:real^3`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` v:real^3`;` v1:real^3`;` w:real^3`;`w1:real^3`];
 DISJ_CASES_TAC(SET_RULE`(w1=v)\/ ~(w1=v:real^3)`)
THENL[
FIND_ASSUM MP_TAC`{v1,w1} IN (E:(real^3->bool)->bool)`
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`{v,u}={u,v}`]
THEN STRIP_TAC
THEN MRESA_TAC fan_run_in_small2_is_fan [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`w1:real^3`;`u:real^3`;` w:real^3`;`v1:real^3`]
THEN EXISTS_TAC`t1:real`
THEN ASM_REWRITE_TAC[lem];
DISJ_CASES_TAC(SET_RULE`(w1=u)\/ ~(w1=u:real^3)`)
THENL[
FIND_ASSUM MP_TAC`{v1,w1} IN (E:(real^3->bool)->bool)`
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`{v,u}={u,v}`]
THEN STRIP_TAC
THEN MRESA_TAC fan_run_in_small3_is_fan[`x:real^3`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` v:real^3`;` w1:real^3`;` w:real^3`;`v1:real^3`]
THEN EXISTS_TAC`t1:real`
THEN ASM_REWRITE_TAC[lem];
SUBGOAL_THEN`{v,u} INTER {v1,w1:real^3}={}` ASSUME_TAC
THENL[
ASM_TAC
THEN SET_TAC[];
MRESA_TAC fan_run_in_small1_is_fan[`x:real^3`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` v:real^3`;` u:real^3`;` w:real^3`;`v1:real^3`;`w1:real^3`]]]]]]));;








let fan_run1_in_small_is_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) (E':(real^3->bool)->bool) v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ E' SUBSET E
/\ (&0< azim x u w v ) /\ (azim x u w v <pi)
/\ sigma_fan x V E u w = v
==>
?h:real. &0<h /\ h<= &1
/\ (!s:real. &0< s /\ s< h==> aff_gt{x} {v,(&1-s)%u+ s % w} INTER {v | ?e. e IN E' /\ v IN aff_ge {x} e}={})`,

REPEAT STRIP_TAC 
THEN MRESA_TAC set_edges_is_finite_fan [`x:real^3`;`V:real^3->bool`;`E:(real^3->bool)->bool`]
THEN MRESA_TAC  FINITE_SUBSET [`(E':(real^3->bool)->bool)`;`(E:(real^3->bool)->bool)`]
THEN ABBREV_TAC`n=CARD (E':(real^3->bool)->bool)`
THEN REPEAT(POP_ASSUM MP_TAC)
THEN SPEC_TAC (`(E':(real^3->bool)->bool)`,`(E':(real^3->bool)->bool)`)
THEN SPEC_TAC (`n:num`,`n:num`)
THEN INDUCT_TAC
THENL(*1*)[
REPEAT STRIP_TAC 
THEN MP_TAC(ISPECL[`E':(real^3->bool)->bool`]CARD_EQ_0)
THEN RESA_TAC
THEN EXISTS_TAC`&1 / &2`
THEN REWRITE_TAC[REAL_ARITH`&0< &1/ &2`;REAL_ARITH`&1 / &2 <= &1`]
THEN ASM_SET_TAC[];(*1*)

REPEAT GEN_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC "A")
THEN REPEAT STRIP_TAC
THEN MP_TAC(ISPEC`(E':(real^3->bool)->bool)` CHOOSE_SUBSET)
THEN RESA_TAC
THEN POP_ASSUM(fun th-> MP_TAC(ISPEC`n:num `th))
THEN REWRITE_TAC[ARITH_RULE `n:num <= SUC n`; HAS_SIZE]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE` t SUBSET E' /\ E' SUBSET E ==> (t:(real^3->bool)->bool) SUBSET E`)
THEN RESA_TAC
THEN REMOVE_THEN "A" (fun th-> MP_TAC(ISPEC`(t:(real^3->bool)->bool)`th))
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"YEU")
THEN SUBGOAL_THEN `~((E':(real^3->bool)->bool) DIFF (t:(real^3->bool)->bool)= {})` ASSUME_TAC
THENL(*2*)[
STRIP_TAC
THEN MP_TAC(SET_RULE`(E':(real^3->bool)->bool) DIFF (t:(real^3->bool)->bool)={} /\ t SUBSET E' ==>  t= E'`)
THEN RESA_TAC
THEN FIND_ASSUM MP_TAC`CARD (t:(real^3->bool)->bool)=n`
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN ASM_REWRITE_TAC[]
THEN ARITH_TAC;(*2*)
SUBGOAL_THEN`?e. e IN (E':(real^3->bool)->bool) DIFF (t:(real^3->bool)->bool)` ASSUME_TAC
THENL(*3*)[
ASM_SET_TAC[];(*3*)

POP_ASSUM MP_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`e IN (E':(real^3->bool)->bool) DIFF (t:(real^3->bool)->bool)/\
(E':(real^3->bool)->bool) SUBSET (E:(real^3->bool)->bool) /\ t SUBSET E' ==> e IN E'/\ e IN E/\ ~(e IN t) /\ {e} UNION t SUBSET E'`)
THEN RESA_TAC
THEN MP_TAC(ISPECL [`{e:(real^3->bool)} UNION (t:(real^3->bool)->bool)`;`(E':(real^3->bool)->bool)`] FINITE_SUBSET)
THEN RESA_TAC
THEN ASSUME_TAC(SET_RULE`e IN {e:(real^3->bool)} UNION (t:(real^3->bool)->bool)`)
THEN MP_TAC(ISPECL[`e:real^3->bool`;`{e:(real^3->bool)} UNION (t:(real^3->bool)->bool)`;]CARD_DELETE)
THEN RESA_TAC
THEN MP_TAC(SET_RULE `e IN {e:(real^3->bool)} UNION (t:(real^3->bool)->bool) 
==> ({e:(real^3->bool)} UNION (t:(real^3->bool)->bool)) DELETE e PSUBSET {e:(real^3->bool)} UNION (t:(real^3->bool)->bool)`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`({e:(real^3->bool)} UNION (t:(real^3->bool)->bool))DELETE e`;`{e:(real^3->bool)} UNION (t:(real^3->bool)->bool)`]CARD_PSUBSET)
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN FIND_ASSUM MP_TAC`FINITE ( {e:(real^3->bool)} UNION (t:(real^3->bool)->bool))`
THEN DISCH_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN DISCH_TAC
THEN MP_TAC(ARITH_RULE`CARD (({e:(real^3->bool)} UNION (t:(real^3->bool)->bool)) DELETE e) < CARD ( {e:(real^3->bool)} UNION (t:(real^3->bool)->bool))
/\ CARD (({e:(real^3->bool)} UNION (t:(real^3->bool)->bool)) DELETE e) = CARD ({e:(real^3->bool)} UNION (t:(real^3->bool)->bool))-1
<=>CARD (({e:(real^3->bool)} UNION (t:(real^3->bool)->bool)) DELETE e) +1= CARD ({e:(real^3->bool)} UNION (t:(real^3->bool)->bool))`)
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th;])
THEN REWRITE_TAC[ARITH_RULE`A=A`; ]
THEN MP_TAC(SET_RULE`~(e IN t)==>({e:(real^3->bool)} UNION (t:(real^3->bool)->bool)) DELETE e=t`)
THEN RESA_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN FIND_ASSUM MP_TAC`(CARD (E':(real^3->bool)->bool)=SUC n)`
THEN REWRITE_TAC[ARITH_RULE`SUC n=(n:num) +1`]
THEN DISCH_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)])
THEN DISCH_TAC
THEN MP_TAC(ISPECL[`{e:(real^3->bool)} UNION (t:(real^3->bool)->bool)`;`E':(real^3->bool)->bool`]CARD_SUBSET_EQ)
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)])
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"MA")
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool) `;`(E:(real^3->bool)->bool)`;`(e:real^3->bool)`]expand_edge_graph_fan)
THEN RESA_TAC
THEN MRESA_TAC fan_run_in_small_is_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;` u:real^3`;`w:real^3`;`v':real^3`;`w':real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th)] THEN ASSUME_TAC(th))
THEN POP_ASSUM MP_TAC
THEN RESA_TAC
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"BE")
THEN EXISTS_TAC` min (h:real) (t1:real)`
THEN STRIP_TAC
THENL(*4*)[
ASM_TAC
THEN REAL_ARITH_TAC;(*4*)

STRIP_TAC
THENL(*5*)[
ASM_TAC
THEN REAL_ARITH_TAC;(*5*)
REPEAT STRIP_TAC
THEN REMOVE_THEN "MA" MP_TAC
THEN DISCH_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th)] THEN ASSUME_TAC(th))
THEN REWRITE_TAC[UNION;IN_ELIM_THM;EXTENSION; INTER;]
THEN GEN_TAC
THEN EQ_TAC
THENL(*6*)[
ASM_REWRITE_TAC[IN_SING]
THEN MP_TAC(REAL_ARITH`s< min h t1==> s<h /\ s<t1`)
THEN RESA_TAC
THEN STRIP_TAC
THENL(*7*)[


POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN REMOVE_THEN "BE"(fun th-> MRESA1_TAC th `s:real`)
THEN ASM_SET_TAC[];

POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN REMOVE_THEN "YEU"(fun th-> MRESA1_TAC th `s:real`)
THEN ASM_SET_TAC[]];
SET_TAC[]]]]]]]);;





let fan_run_in_small_is_not_meet_xfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ (&0< azim x u w v ) /\ (azim x u w v <pi)
/\ sigma_fan x V E u w = v
==>
?h:real. &0<h /\ h<= &1
/\ (!s:real. &0< s /\ s< h==> aff_gt{x} {v,(&1-s)%u+ s % w} INTER {v | ?e. e IN E /\ v IN aff_ge {x} e}={})`,

REPEAT STRIP_TAC
THEN MP_TAC(ISPECL[`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(E:(real^3->bool)->bool)`;` (v:real^3)`;`u:real^3`;` (w:real^3)`]fan_run1_in_small_is_fan)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[SET_RULE`A SUBSET A`]
THEN RESA_TAC
THEN EXISTS_TAC`h:real`
THEN ASM_REWRITE_TAC[]
THEN ASM_SET_TAC[]);;
let fan_run_in_small_is_subset_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ (&0< azim x u w v ) /\ (azim x u w v <pi)
/\ sigma_fan x V E u w = v
==>
    ?h:real. &0<h /\ h<= &1
/\ (!s:real. &0< s /\ s< h==> aff_gt{x} {v,(&1-s)%u+ s % w} SUBSET yfan(x,V,E))`,
REPEAT STRIP_TAC
THEN MP_TAC(ISPECL[`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(E:(real^3->bool)->bool)`;` (v:real^3)`;`u:real^3`;` (w:real^3)`]fan_run1_in_small_is_fan)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[SET_RULE`A SUBSET A`;yfan;xfan]
THEN RESA_TAC
THEN EXISTS_TAC`h:real`
THEN ASM_REWRITE_TAC[]
THEN ASM_SET_TAC[]);;


let not_collinear_is_properties_fully_surrounded1=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 t:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ (&0< azim x u w v ) /\ (azim x u w v <pi)
/\  &0<=t /\ t<= &1
==> ~ collinear {x,v,(&1-t)%u+ t % w}`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[]
THEN MP_TAC(REAL_ARITH`&0<=t /\ t<= &1 ==> t=  &0 \/ t= &1 \/ (&0<t /\ t< &1)`)
THEN RESA_TAC
THENL[
REDUCE_ARITH_TAC THEN REDUCE_VECTOR_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`] ;

REWRITE_TAC[REAL_ARITH`&1- &1= &0`] THEN REDUCE_VECTOR_TAC
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC NOT_COPLANAR_NOT_COLLINEAR[`x:real^3`;`v:real^3`;`w:real^3`;`u:real^3`]
THEN POP_ASSUM MATCH_MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,B,D,C}`]
THEN ASM_REWRITE_TAC[];

MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`] 
THEN MRESA_TAC collinear1_fan [`x:real^3`;`v:real^3`;`(&1-t)% u+t% w:real^3`]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC properties_fully_surrounded[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;` w:real^3`]
THEN POP_ASSUM MP_TAC
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff; AFFINE_HULL_2; IN_ELIM_THM; coplanar]
THEN STRIP_TAC
THEN EXISTS_TAC`x:real^3`
THEN EXISTS_TAC`v:real^3`
THEN EXISTS_TAC`u:real^3`
THEN SUBGOAL_THEN`(x:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&0`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;

SUBGOAL_THEN`(v:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC`&0`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;

SUBGOAL_THEN`(u:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL[ 
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;

SUBGOAL_THEN`(w:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(REAL_ARITH`&0<t:real==> ~(t= &0)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(t:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - t) % u + t % w = u' % x + v' % v
<=>t % w = u' % x + v' % v-(&1 - t) % u :real^3`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t % w = u' % x + v' % v-(&1 - t) % u 
==> (inv (t))%(t % w  ) = (inv (t))%(u' % x + v' % v-(&1 - t) % u ):real^3`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%(B+C-(&1-E)%D)=A%B+A%C+(A*E-A)%D`;VECTOR_ARITH`A%B%C=(A*B)%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`(inv t * u'):real`
THEN EXISTS_TAC`(inv t * v'):real`
THEN EXISTS_TAC`(&1-inv t):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`A*B+A*C+D-E=A*(B+C)+D-E`]
THEN ASM_TAC
THEN REAL_ARITH_TAC;

ASM_TAC
THEN SET_TAC[]]]]]]);;



let not_collinear_is_properties_fully_surrounded=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 t:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ (&0< azim x u w v ) /\ (azim x u w v <pi)
/\  &0<t /\ t< &1
==> ~ collinear {x,v,(&1-t)%u+ t % w}`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[]
THEN MP_TAC(REAL_ARITH`&0 <t /\ t < &1==> &0<= t /\ t <= &1`)
THEN RESA_TAC
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded1 [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`t:real`]
);;

let exists_inf_element_fix_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3  u1:real^3 .
FAN(x,V,E)/\ v IN V  /\ CARD (set_of_edge v V E) >1
==>
(?(u:real^3). (u IN (set_of_edge v V E)) /\
(!(w:real^3). (w IN (set_of_edge v V E)) ==> azim x v u1 u <= azim x v u1 w))`,
(let lemma = prove
   (`!X:real->bool. 
          FINITE X /\ ~(X = {}) 
          ==> ?a. a IN X /\ !x. x IN X ==> a <= x`,
    MESON_TAC[INF_FINITE]) in
MP_TAC(lemma) 
THEN DISCH_THEN(LABEL_TAC "a") 
THEN REPEAT GEN_TAC THEN REWRITE_TAC[FAN;fan1] 
THEN STRIP_TAC 
THEN MRESA_TAC remark_finite_fan1[`(v:real^3)` ;`(V:real^3->bool)`; `(E:(real^3->bool)->bool)`]
THEN DISJ_CASES_TAC(SET_RULE`((set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool))={})\/
 ~((set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)) ={})`)
THENL[
FIND_ASSUM MP_TAC `CARD ((set_of_edge v V E):real^3->bool) > 1`
THEN POP_ASSUM (fun th->REWRITE_TAC[th;CARD_CLAUSES])
THEN ARITH_TAC;
SUBGOAL_THEN`~(IMAGE (azim x v u1) ((set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)) )={})` ASSUME_TAC
THENL[
REWRITE_TAC[IMAGE_EQ_EMPTY] THEN ASM_MESON_TAC[];

SUBGOAL_THEN` FINITE (IMAGE (azim x v u1) (set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)))` ASSUME_TAC
THENL[ASM_MESON_TAC[FINITE_IMAGE];

REMOVE_THEN "a" (fun th ->MRESAL1_TAC th `(IMAGE (azim x v u1) ((set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)) ))`[IMAGE;IN_ELIM_THM]) 
THEN EXISTS_TAC`x':real^3`
  THEN ASM_REWRITE_TAC[] 
THEN ASM_MESON_TAC[]]]]));;


let exists_element_in_half_sapace_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3  u1:real^3 w1:real^3.
FAN(x,V,E)/\ v IN V /\ ~coplanar{x,v,u1,w1} 
/\ CARD (set_of_edge v V E) >1
/\ fan80(x,V,E) 
==>  ?u:real^3. {v,u} IN E /\ &0< azim x v u1 (u:real^3)/\ azim x v u1 (u:real^3) <pi  `,
REWRITE_TAC[fan80]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "BE")
THEN MRESA_TAC NOT_COPLANAR_NOT_COLLINEAR [`x:real^3`;`v:real^3`;`u1:real^3`;`w1:real^3`]
THEN MRESA_TAC exists_inf_element_fix_fan [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u1:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "YEU")
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` u:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC
THEN RESA_TAC
THEN MRESA_TAC SUR_SIGMA_FAN[`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(v:real^3) `;`(u:real^3)`]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`v:real^3`]
THEN DISJ_CASES_TAC(REAL_ARITH`(pi<= azim (x:real^3) (v:real^3) (u1:real^3) u  )\/  azim x v u1 (u:real^3) <pi `)
THENL[
REMOVE_THEN "BE" (fun th-> MRESA_TAC th [`v:real^3`;`w:real^3`])
THEN MP_TAC(REAL_ARITH`pi <= azim (x:real^3) (v:real^3) (u1:real^3) (u:real^3) /\ azim x v w u < pi==> azim x v w u <= azim x v u1 u`)
THEN RESA_TAC
THEN MRESA_TAC sum5_azim_fan[`x:real^3`;`v:real^3`;`u1:real^3`;`w:real^3`;`u:real^3`]
THEN MP_TAC(REAL_ARITH`&0 < azim x v w u /\ azim (x:real^3) (v:real^3) (u1:real^3) (u:real^3) = azim x v u1 w + azim x v w u
==> azim x v u1 w < azim x v u1 u`)
THEN POP_ASSUM (fun th-> GEN_REWRITE_TAC( LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN RESA_TAC
THEN REMOVE_THEN "YEU" (fun th-> MRESA1_TAC th `w:real^3`)
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;

ASSUME_TAC(ISPECL[`(x:real^3)`;`(v:real^3)`;`(u1:real^3)`;`u:real^3`]azim)
THEN MP_TAC(REAL_ARITH`(&0<= azim (x:real^3) (v:real^3) (u1:real^3) u) ==>(azim (x:real^3) (v:real^3) (u1:real^3) u = &0 ) \/  &0< azim x v u1 (u:real^3)  `)
THEN POP_ASSUM (fun th-> GEN_REWRITE_TAC( LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th] THEN REWRITE_TAC[])
THEN STRIP_TAC
THENL[
EXISTS_TAC`sigma_fan x V E v (u:real^3)`
THEN DISJ_CASES_TAC(SET_RULE`(set_of_edge v V E = {u:real^3})\/  ~(set_of_edge v V E = {u})`)
THENL[

MRESA_TAC CARD_SING[`u:real^3`; `(set_of_edge v V E):real^3->bool`]
THEN FIND_ASSUM MP_TAC `CARD ((set_of_edge v V E):real^3->bool) >1`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun TH-> REWRITE_TAC[TH])
THEN ARITH_TAC;

MRESA_TAC SIGMA_FAN[`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(v:real^3) `;`(u:real^3)`]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;`sigma_fan x V E v (u:real^3)`;`v:real^3`]
THEN POP_ASSUM MP_TAC
THEN RESA_TAC
THEN REMOVE_THEN "YEU" (fun th-> MRESA1_TAC th `sigma_fan x V E v (u:real^3)`)
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`v:real^3`;`u1:real^3`;`u:real^3`;`sigma_fan x V E v (u:real^3)`]
THEN REDUCE_ARITH_TAC
THEN REMOVE_THEN "BE" (fun th-> MRESA_TAC th [`v:real^3`;`(u:real^3)`])];

EXISTS_TAC`(u:real^3)`
THEN ASM_REWRITE_TAC[]]]);;






let JBDNJJB=prove(`!u:real^3 v:real^3 w:real^3.
~collinear {vec 0, u, v} /\ ~collinear {vec 0, u, w}
==>
?t:real. &0< t /\ sin(azim (vec 0) u v w)=t *(u cross v) dot w`,

REPEAT STRIP_TAC
THEN MRESA_TAC th3[`((vec 0):real^3)`;` (u:real^3)`;` (v:real^3)`]
THEN MRESA_TAC properties_coordinate[`((vec 0):real^3)`;` (u:real^3)`;` (v:real^3)`]
THEN MRESA_TAC azim[`((vec 0):real^3)`;` (u:real^3)`;` (v:real^3)`;`(w:real^3)`]
THEN POP_ASSUM (fun th->MRESA_TAC th [`e1_fan ((vec 0):real^3) (u:real^3) (v:real^3)`;`e2_fan ((vec 0):real^3) (u:real^3) (v:real^3)`;`e3_fan ((vec 0):real^3) (u:real^3) (v:real^3)`])
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"YEU EM")
THEN DISCH_TAC THEN DISCH_TAC
THEN MRESA_TAC sincos1_of_u_fan[`((vec 0):real^3)`;` (u:real^3)`;` (v:real^3)`;`r1:real`; `psi:real`; `h1:real`]
THEN REMOVE_THEN "YEU EM" MP_TAC
THEN ASM_REWRITE_TAC[COS_ADD;SIN_ADD;]
THEN REDUCE_ARITH_TAC
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`w =
      (r2 * cos (azim (vec 0) u v w)) % e1_fan (vec 0) u v +
      (r2 * sin (azim (vec 0) u v w)) % e2_fan (vec 0) u v +
      h2 % u ==>
(u cross v) dot w =
(u cross v) dot ((r2 * cos (azim (vec 0) u v w)) % e1_fan (vec 0) u v +
      (r2 * sin (azim (vec 0) u v w)) % e2_fan (vec 0) u v +
      h2 % u)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN REWRITE_TAC[DOT_LADD; DOT_RADD; DOT_LMUL; DOT_RMUL;DOT_CROSS_SELF; e2_fan;e1_fan;e3_fan;
VECTOR_ARITH`A- vec 0= A`;CROSS_LADD; CROSS_RADD; CROSS_LMUL; CROSS_RMUL;CROSS_REFL;CROSS_RNEG;CROSS_LNEG;]
THEN REDUCE_ARITH_TAC
THEN REWRITE_TAC[NORM_MUL;REAL_INV_MUL; REAL_ABS_INV;REAL_INV_INV;REAL_ABS_NORM;DOT_SQUARE_NORM
;REAL_ARITH`(r2 * sin (azim (vec 0) u v w)) *
 (norm u * inv (norm (u cross v))) *
 inv (norm u) *
 norm (u cross v) pow 2 =(r2* norm(u cross v)) * sin (azim (vec 0) u v w) *
 ( inv (norm u) * norm u)*
( inv (norm (u cross v))* norm (u cross (v:real^3)))`
]
THEN MP_TAC(ISPECL[`u:real^3`;`(vec 0) :real^3`]imp_norm_not_zero_fan)
THEN REDUCE_VECTOR_TAC
THEN RESA_TAC
THEN MP_TAC(ISPEC`(norm(u:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`u:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(&0 =norm(u:real^3)) /\ &0 <= norm(u:real^3)==> &0 <norm(u:real^3)`)
THEN RESA_TAC
THEN SUBGOAL_THEN`~(u cross v = vec 0)` ASSUME_TAC
THENL[ASM_REWRITE_TAC[CROSS_EQ_0];

MP_TAC(ISPECL[`u cross v :real^3`;`(vec 0) :real^3`]imp_norm_not_zero_fan)
THEN REDUCE_VECTOR_TAC
THEN RESA_TAC
THEN MP_TAC(ISPEC`(norm(u cross v:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`u cross v:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(&0 =norm(u cross v:real^3)) /\ &0 <= norm(u cross v:real^3)==> &0 <norm(u cross v:real^3)`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`r2:real`;`norm(u cross v:real^3)`]
THEN MP_TAC(REAL_ARITH`&0<(r2:real)*norm(u cross v:real^3)==> ~((r2:real)*norm(u cross v:real^3)= &0)`)
THEN REDUCE_VECTOR_TAC
THEN RESA_TAC
THEN MP_TAC(ISPEC`(r2 * norm(u cross v:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(r2 * norm(u cross v:real^3))`REAL_LT_INV)
THEN RESA_TAC
THEN REDUCE_ARITH_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(u cross v) dot w = (r2 * norm (u cross v)) * sin (azim (vec 0) u v w) ==>
inv (r2 * norm (u cross v))*(r2 * norm (u cross v)) * sin (azim (vec 0) u v w)= inv (r2 * norm (u cross v)) *((u cross v) dot w)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (r2 * norm (u cross v)) * (r2 * norm (u cross v)) *
 sin (azim (vec 0) u v w)=(inv (r2 * norm (u cross v)) * (r2 * norm (u cross v)))*
 sin (azim (vec 0) u v w)`]
THEN REDUCE_ARITH_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`inv (r2 * norm (u cross v)):real`
THEN ASM_REWRITE_TAC[]]);;

let independent_run_edges_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ fan80(x,V,E)
/\ sigma_fan x V E u w = v
/\ &0<a /\ a<= &1
==> independent {v - x, u - x, ((&1 - a) % u + a % w) - x}`,


REPEAT STRIP_TAC
THEN FIND_ASSUM MP_TAC `fan80(x:real^3,V,E)`
THEN REWRITE_TAC[fan80]
THEN DISCH_TAC
THEN POP_ASSUM (fun th -> MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC "YEU EM"))
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MP_TAC(REAL_ARITH`&0< a==> ~(a= &0)`)
THEN RESA_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (u:real^3)`;`v:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] 
THEN MRESA_TAC continuous_coplanar_fan[`x:real^3 `;` v:real^3`;` u:real^3`;` w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `a:real`)
THEN POP_ASSUM MP_TAC
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3, v, u, (&1 - a) % u + a % w}= {(--x) +x, (--x) +v, (--x) +u, (--x) +(&1 - a) % u + a % w}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3,v:real^3,u:real^3,(&1 - a) % u + a % w:real^3}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN REMOVE_ASSUM_TAC
THEN DISCH_TAC
THEN MRESA_TAC NOT_COPLANAR_0_4_IMP_INDEPENDENT[`v - x:real^3`; `u - x:real^3`; `((&1 - a) % u + a % w) - x:real^3`]);;

let span_run_edges_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ fan80(x,V,E)
/\ sigma_fan x V E u w = v
/\ &0<a /\ a< &1
==> ?t1:real t2:real t3:real. 
u1-x =t1 % (v-x)+t2 % ((&1 - a) % u + a % w - x)+t3 %(u-x)`,
REPEAT STRIP_TAC
THEN FIND_ASSUM MP_TAC `fan80(x:real^3,V,E)`
THEN REWRITE_TAC[fan80]
THEN DISCH_TAC
THEN POP_ASSUM (fun th -> MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC "YEU EM"))
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MP_TAC(REAL_ARITH`&0< a==> ~(a= &0)`)
THEN RESA_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (u:real^3)`;`v:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] 
THEN MRESA_TAC continuous_coplanar_fan[`x:real^3 `;` v:real^3`;` u:real^3`;` w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `a:real`)
THEN POP_ASSUM MP_TAC
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3, v, u, (&1 - a) % u + a % w}= {(--x) +x, (--x) +v, (--x) +u, (--x) +(&1 - a) % u + a % w}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3,v:real^3,u:real^3,(&1 - a) % u + a % w:real^3}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN REMOVE_ASSUM_TAC
THEN DISCH_TAC
THEN MRESA_TAC NOT_COPLANAR_0_4_IMP_INDEPENDENT[`v - x:real^3`; `u - x:real^3`; `((&1 - a) % u + a % w) - x:real^3`]
THEN MRESA_TAC NOT_COPLANAR_NOT_COLLINEAR[`v - x:real^3`; `u - x:real^3`; `((&1 - a) % u + a % w) - x:real^3`;`(vec 0):real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={D,A,B,C}`]
THEN RESA_TAC
THEN POP_ASSUM(fun th-> MP_TAC(th) THEN ASSUME_TAC(th))
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,D}={D,A,B}`] 
THEN DISCH_TAC
THEN MRESA_TAC th3[`v - x:real^3`; `u - x:real^3`; `((&1 - a) % u + a % w) - x:real^3`]
THEN MRESA_TAC th3[ `((&1 - a) % u + a % w) - x:real^3`;`v - x:real^3`; `u - x:real^3`;]
THEN MP_TAC(ISPECL [`(:real^3)`; `{v - x, u - x, ((&1 - a) % u + a % w) - x:real^3}`] CARD_EQ_DIM) THEN
  ASM_SIMP_TAC[ORTHONORMAL_IMP_INDEPENDENT; SUBSET_UNIV] THEN
  REWRITE_TAC[DIM_UNIV; DIMINDEX_3; HAS_SIZE; FINITE_INSERT; FINITE_EMPTY] THEN
  SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY; IN_INSERT;] 
THEN  ASM_REWRITE_TAC[NOT_IN_EMPTY; ARITH;SUBSET] 
THEN DISCH_TAC
THEN POP_ASSUM(fun th-> MRESAL1_TAC th `u1-x:real^3`[IN_UNIV;SPAN_3;IN_ELIM_THM])
THEN EXISTS_TAC`u':real`	
THEN EXISTS_TAC`w':real`
THEN EXISTS_TAC`v':real`
THEN VECTOR_ARITH_TAC);;


let IMP_NORM_FAN=prove(`!va:real^3 vb:real^3. ~(va = vb)
==> ~(norm (va-vb) = &0) /\ &0 <= norm (va-vb) /\ &0 < norm (va-vb) /\ &0 <= inv (norm (va-vb)) 
/\ &0 < inv (norm (va-vb)) /\ inv (norm (va-vb)) * norm (va-vb) = &1`,
REPEAT GEN_TAC
THEN DISCH_TAC
THEN MRESA_TAC imp_norm_not_zero_fan[`va:real^3`;`vb:real^3`]
THEN ASSUME_TAC(ISPEC`va-vb:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(norm(va-vb:real^3)= &0) /\ &0 <= norm(va-vb:real^3)==> &0 <norm(va-vb:real^3)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LE_INV `norm(va-vb:real^3)`
THEN MRESA1_TAC REAL_LT_INV `norm(va-vb:real^3)`
THEN MRESA1_TAC REAL_MUL_LINV `norm(va-vb:real^3)`
THEN ASM_REWRITE_TAC[]);;



let cross_dot_fully_surrounded_fan=prove(`!x:real^3 v1:real^3 u1:real^3 v:real^3.
~collinear{x,v1,u1}
/\ ~collinear{x,v1,v}
/\ &0< azim x v1 v u1
/\  azim x v1 v u1 < pi
==> &0 < ((v1 - x) cross (v - x)) dot (u1 - x)`,

REPEAT STRIP_TAC
THEN MRESA1_TAC SIN_POS_PI`azim x v1 v (u1:real^3)`
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC AZIM_TRANSLATION[`-- x:real^3`;`x:real^3`;`v1:real^3`;` v:real^3`;`u1:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`x-x= vec 0`;VECTOR_ARITH`(-- X)+A= A-X:real^3`]
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN MRESA_TAC JBDNJJB[`(v1-x):real^3`;`v-x:real^3`;`u1-x:real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM COLLINEAR_3;]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN RESA_TAC
THEN MRESAL_TAC REAL_LT_LMUL_EQ [` &0:real `;`(((v1 - x) cross (v - x)) dot ((u1 - x):real^3)):real`;`t:real`][REAL_ARITH`a * &0 = &0`]);;

let cross_dot_fully_surrounded_ge_fan=prove(`!x:real^3 v1:real^3 u1:real^3 v:real^3.
~collinear{x,v1,u1}
/\ ~collinear{x,v1,v}
/\ &0<= azim x v1 v u1
/\  azim x v1 v u1 <= pi
==> &0 <= ((v1 - x) cross (v - x)) dot (u1 - x)`,
REPEAT STRIP_TAC
THEN MRESA1_TAC SIN_POS_PI_LE`azim x v1 v (u1:real^3)`
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC AZIM_TRANSLATION[`-- x:real^3`;`x:real^3`;`v1:real^3`;` v:real^3`;`u1:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`x-x= vec 0`;VECTOR_ARITH`(-- X)+A= A-X:real^3`]
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN MRESA_TAC JBDNJJB[`(v1-x):real^3`;`v-x:real^3`;`u1-x:real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM COLLINEAR_3;]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN RESA_TAC
THEN MRESAL_TAC REAL_LE_LMUL_EQ [` &0:real `;`(((v1 - x) cross (v - x)) dot ((u1 - x):real^3)):real`;`t:real`][REAL_ARITH`a * &0 = &0`]);;

let AFF_LT_2_1 = prove
 (`!x v w.
        DISJOINT {x,v} {w}
        ==> aff_lt {x,v} {w} =
             {y | ?t1 t2 t3.
                      t3 < &0 /\
                     t1 + t2 + t3 = &1 /\
                     y = t1 % x + t2 % v + t3 % w}`,
  AFF_TAC);;


let properties_of_collinear4_points_fan=prove(`!x:real^3  v:real^3 u:real^3 v1:real^3.
~collinear{x,v,u}
/\ v1 IN aff_gt {x} {v,u}
==> ~collinear{x,v1,v}`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(u:real^3) `;]
THEN MRESA_TAC AFF_GT_1_2[`(x:real^3)` ;` (v:real^3)`;`(u:real^3) `;]
THEN POP_ASSUM(fun th-> REWRITE_TAC[th;IN_ELIM_THM])
THEN STRIP_TAC 
THEN ASM_REWRITE_TAC[collinear1_fan;]
THEN FIND_ASSUM MP_TAC`~(u IN aff {x, v:real^3})`
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1 % x + t2 % v + t3 % u = u' % x + v' % v <=> t3 % u = (u'-t1) % x + (v'-t2) % v`;]
THEN MP_TAC (REAL_ARITH`&0< t3==> ~(t3= &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV `(t3:real)`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`t3 % u = (u'-t1) % x + (v'-t2) % v ==> (inv (t3)) % (t3) % ( u) = (inv (t3)) % ( (u'-t1) % x + (v'-t2) % v:real^3)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th] 
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(A%B%C=(A*B)%C:real^3)`;VECTOR_ARITH`A%(B+C)=A%B+A%C`])
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`inv t3 * ((u' - t1):real)`
THEN EXISTS_TAC`inv t3 * ((v' - t2):real)`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv t3 * (u' - t1) + inv t3 * (v' - t2)=inv t3 *(t3+ (u'+v') -( t1+ t2+t3))`;REAL_ARITH`A+ &1- &1=A`]);;



let cross_dot_fully_surrounded1_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real v1:real^3 u1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<a /\ a< &1
/\ fan80(x,V,E)
/\ ~collinear{x,v1,u1}
/\ v1 IN aff_gt {x} {v,(&1-a)%u+ a%w}
/\ &0< azim x v1 v u1
/\  azim x v1 v u1 < pi
==> &0< ((v1 - x) cross (u1 - x)) dot
  ((&1 - a) % u + a % w-x)`,

REPEAT STRIP_TAC
THEN MRESAL_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`v1:real^3`; `(&1 - a) % u + a % w:real^3`;`u1:real^3`][VECTOR_ARITH`((&1 - a) % u + a % w) - x=(&1 - a) % u + a % w - x`]
THEN POP_ASSUM MATCH_MP_TAC
THEN FIND_ASSUM MP_TAC `fan80(x:real^3,V,E)`
THEN REWRITE_TAC[fan80]
THEN DISCH_THEN(LABEL_TAC"yeu em")
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (u:real^3)`;`v:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] 
THEN REMOVE_THEN "yeu em" (fun th -> MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC "YEU EM"))
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`]
THEN MRESA_TAC properties_of_collinear4_points_fan[`x:real^3`;`v:real^3`;`(&1-a)%u+ a%w:real^3`;`v1:real^3`]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - a) % (u:real^3) + a % (w:real^3)`;]
THEN MRESA_TAC AFF_GT_1_2[`(x:real^3)` ;` (v:real^3)`;`(&1 - a) % (u:real^3) + a % (w:real^3) `;]
THEN FIND_ASSUM MP_TAC `(v1:real^3) IN aff_gt {x} {v,(&1-a)%u+ a%(w:real^3)}`
THEN POP_ASSUM(fun th-> REWRITE_TAC[th;IN_ELIM_THM])
THEN STRIP_TAC 
THEN SUBGOAL_THEN`~collinear {x, v1, (&1 - a) % u + a % w:real^3}` ASSUME_TAC
THENL[
ASM_REWRITE_TAC[collinear1_fan;]
THEN FIND_ASSUM MP_TAC`~((&1 - a) % u + a % w IN aff {x, v:real^3})`
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1 % x + t2 % v + t3 % ((&1 - a) % u + a % w) =
      u' % x + v' % ((&1 - a) % u + a % w)<=>
(&1-(t1+t2+t3)) % x  + (v'-t3) % ((&1 - a) % u + a % w-x) =
      (&1-(u'+v') ) % x + t2 % (v-x)`;REAL_ARITH`a-a= &0`]
THEN REDUCE_VECTOR_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`v' - t3= &0 \/ ~(v' - (t3:real)= &0)`)
THENL[
ASM_REWRITE_TAC[VECTOR_ARITH`&0 %A=B<=> B= vec 0`;VECTOR_MUL_EQ_0]
THEN MP_TAC(REAL_ARITH`&0< t2==> ~(t2 = &0)`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A-B= vec 0<=> A=B`];

MRESA1_TAC REAL_MUL_LINV `(v'-t3:real)`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(v' - t3) % ((&1 - a) % u + a % w - x) = t2 % (v - x:real^3) ==> (inv (v'-t3)) % (v' - t3) % ((&1 - a) % u + a % w - x) = (inv (v'-t3)) % ( t2 % (v - x))`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th] 
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(A%B%C=(A*B)%C:real^3)`])
THEN REWRITE_TAC[VECTOR_ARITH`&1 % ((&1 - a) % u + a % w - x) = (inv (v' - t3) * t2) % (v - x)<=>
(&1 - a) % u + a % w  = (&1-(inv (v' - t3) * t2))%x +(inv (v' - t3) * t2) % v:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`&1-(inv (v' - t3) * (t2:real))`
THEN EXISTS_TAC`(inv (v' - t3) * (t2:real))`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC];

POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"A")
THEN DISCH_TAC
THEN REMOVE_THEN "A" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`v1:real^3`;`v:real^3`;`u1:real^3`;`((&1 - a) % u + a % w):real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"B")
THEN DISCH_TAC
THEN SUBGOAL_THEN`azim x v1 v ((&1 - a) % u + a % w)= pi` ASSUME_TAC
THENL[
POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`(x:real^3)` ;`v1:real^3`;` (v:real^3)`]
THEN MRESA_TAC AFF_LT_2_1[`x:real^3`;`v1:real^3`;`v:real^3`]
THEN MRESAL_TAC AZIM_EQ_PI_ALT[`x:real^3`;`v1:real^3`;`v:real^3`;`((&1 - a) % u + a % w):real^3`][IN_ELIM_THM]
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[VECTOR_ARITH`A=B+C+D <=> D=A-B-C:real^3`]
THEN MP_TAC(REAL_ARITH`&0<t3==> ~(t3= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV `(t3:real)`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`t3 % ((&1 - a) % u + a % w) = v1 - t1 % x - t2 % v ==> (inv (t3)) % t3 % ((&1 - a) % u + a % w)  = (inv (t3)) % ( v1 - t1 % x - t2 % v:real^3)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th] 
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(A%B%C=(A*B)%C:real^3)`])
THEN REWRITE_TAC[VECTOR_ARITH`&1 % ((&1 - a) % u + a % w) = inv t3 % (v1 - t1 % x - t2 % v)
<=> (&1 - a) % u + a % w = (-- inv t3 * t1) % x + (inv t3) % v1 +(-- (inv t3* t2)) % v`]
THEN STRIP_TAC
THEN EXISTS_TAC`(-- inv t3 * t1):real`
THEN EXISTS_TAC`(inv t3):real`
THEN EXISTS_TAC`(-- (inv t3 * t2)):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`--inv t3 * t1 + inv t3 + --(inv t3 * t2)= inv t3 *(t3+ &1-(t1+t2+t3))`;REAL_ARITH`(t3 + &1 - &1)=t3`;REAL_ARITH`--A< &0 <=> &0< A`]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LT_INV
THEN ASM_REWRITE_TAC[];
REMOVE_THEN "B" MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN REMOVE_ASSUM_TAC
THEN MP_TAC(REAL_ARITH`azim x v1 v u1 < pi==> azim x v1 v u1 <= pi`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN FIND_ASSUM MP_TAC`azim x v1 v (u1:real^3) < pi`
THEN FIND_ASSUM MP_TAC`&0< azim x v1 v (u1:real^3) `
THEN REAL_ARITH_TAC]]);;









let exists_cross_dot_fully_surrounded1_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real v1:real^3 u1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<a /\ a< &1
/\ fan80(x,V,E)
/\ ~collinear{x,v1,u1}
/\ v1 IN aff_gt {x} {v,(&1-a)%u+ a%w}
/\ &0< azim x v1 v u1
/\  azim x v1 v u1 < pi
==> ?t.  &0< t/\ t < &1
/\( !h:real. &0< h/\ h<t
==> &0< ((v1 - x) cross (u1 - x)) dot (((&1 - h) % ((&1 - a) % u + a % w) + h % u) - x))`,

REPEAT STRIP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`(((&1 - t) % ((&1 - a) % u + a % w) + t % u) - x)=
 ((&1 - a) % u + a % w - x) + t % (--a) % ( w-u)`;]
THEN ABBREV_TAC`va=(&1 - a) % u + a % w-x:real^3`
THEN REWRITE_TAC[DOT_RADD;DOT_RMUL; REAL_ARITH`&0<A+C*(--B)*D<=> (B*D)*C<A`]
THEN MRESA_TAC cross_dot_fully_surrounded1_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` v:real^3`;` u:real^3`;`w:real^3`;`a:real`;`v1:real^3`;`u1:real^3`]
THEN DISJ_CASES_TAC(REAL_ARITH`(((v1 - x) cross (u1 - x:real^3)) dot (w - u)<= &0 )\/ (&0<((v1 - x) cross (u1 - x)) dot (w - u))`)
THENL[ 
EXISTS_TAC`&1/ &2`
THEN REWRITE_TAC[REAL_ARITH`&0 < &1 / &2 /\ &1 / &2 < &1`]
THEN REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0 < a ==> &0<= (a:real)`) THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0 < h ==> &0<= (h:real)`) THEN RESA_TAC
THEN MRESAL_TAC REAL_LE_LMUL[`a:real`;`((v1 - x) cross (u1 - x)) dot ((w - u):real^3)`;`&0`][REAL_ARITH`(A:real)*  &0= &0`]
THEN MRESAL_TAC REAL_LE_RMUL[`a*((v1 - x) cross (u1 - x)) dot ((w - u):real^3)`;`&0`;`h:real`][REAL_ARITH`&0 * (A:real)= &0`]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MRESA_TAC REAL_LT_MUL[`a:real`;`((v1 - x) cross (u1 - x)) dot ((w - u):real^3)`]
THEN MP_TAC(REAL_ARITH`&0<(a:real)*((v1 - x) cross (u1 - x)) dot ((w - u):real^3)==> ~( (a:real)*((v1 - x) cross (u1 - x)) dot ((w - u):real^3)= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_RINV`(a:real)*((v1 - x) cross (u1 - x)) dot ((w - u):real^3)`
THEN MRESA1_TAC REAL_LT_INV`(a:real)*((v1 - x) cross (u1 - x)) dot ((w - u):real^3)`
THEN MRESA_TAC REAL_LT_MUL[`inv((a:real)*((v1 - x) cross (u1 - x)) dot ((w - u):real^3))`;`((v1 - x) cross (u1 - x)) dot (va:real^3)`]
THEN EXISTS_TAC `min ((inv(a *(((v1 - x) cross (u1 - x)) dot (w - u)))*(((v1 - x) cross (u1 - x)) dot va)) / &2 ) (&1/ &2)`
THEN STRIP_TAC
THENL[POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
STRIP_TAC
THENL[
 REAL_ARITH_TAC;
REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH` h< min ((inv (a * (((v1 - x) cross (u1 - x)) dot (w - u))) * (((v1 - x) cross (u1 - x)) dot va)) /  &2) (&1 / &2)
==>
h< ((inv (a * (((v1 - x) cross (u1 - x)) dot (w - u))) * (((v1 - x) cross (u1 - x)) dot va)) /  &2)`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_LMUL[`(a * (((v1 - x) cross (u1 - x)) dot (w - u:real^3)))`;`h:real`;`((inv (a * (((v1 - x) cross (u1 - x)) dot (w - u))) * (((v1 - x) cross (u1 - x:real^3)) dot va)) /  &2)`]
THEN ASM_TAC
THEN ABBREV_TAC`vb=a * (((v1 - x) cross (u1 - x)) dot (w - u))`
THEN REPEAT DISCH_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`vb * (inv vb * (((v1 - x) cross (u1 - x)) dot va)) / &2=
(vb * inv vb) * (((v1 - x) cross (u1 - x)) dot va)/ &2`]
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC]]]);;

let cross_dot_fully_surrounded2_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real v1:real^3 u1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<a /\ a< &1
/\ fan80(x,V,E)
/\ ~collinear{x,v1,u1}
/\ v1 IN aff_gt {x} {v,(&1-a)%u+ a%w}
/\ &0< azim x v1 v u1
/\  azim x v1 v u1 < pi
==> &0 < (((&1 - a) % u + a % w - x) cross (v - x)) dot (u1 - x)`,

REPEAT STRIP_TAC
THEN FIND_ASSUM MP_TAC `fan80(x:real^3,V,E)`
THEN REWRITE_TAC[fan80]
THEN DISCH_THEN(LABEL_TAC"yeu em")
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (u:real^3)`;`v:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] 
THEN REMOVE_THEN "yeu em" (fun th -> MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC "YEU EM"))
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`]
THEN MRESA_TAC properties_of_collinear4_points_fan[`x:real^3`;`v:real^3`;`(&1-a)%u+ a%w:real^3`;`v1:real^3`]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - a) % (u:real^3) + a % (w:real^3)`;]
THEN MRESA1_TAC SIN_POS_PI`azim x v1 v (u1:real^3)`
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC AZIM_TRANSLATION[`-- x:real^3`;`x:real^3`;`v1:real^3`;` v:real^3`;`u1:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`x-x= vec 0`;VECTOR_ARITH`(-- X)+A= A-X:real^3`]
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN MRESA_TAC JBDNJJB[`(v1-x):real^3`;`v-x:real^3`;`u1-x:real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM COLLINEAR_3;]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN RESA_TAC
THEN MRESAL_TAC REAL_LT_LMUL_EQ [` &0:real `;`(((v1 - x) cross (v - x)) dot ((u1 - x):real^3)):real`;`t:real`][REAL_ARITH`a * &0 = &0`]
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN MRESA_TAC AFF_GT_1_2[`(x:real^3)` ;` (v:real^3)`;`(&1 - a) % (u:real^3) + a % (w:real^3) `;]
THEN FIND_ASSUM MP_TAC `v1 IN aff_gt {x} {v,(&1-a)%u+ a%(w:real^3)}`
THEN POP_ASSUM(fun th-> REWRITE_TAC[th;IN_ELIM_THM])
THEN STRIP_TAC 
THEN POP_ASSUM (fun th-> REWRITE_TAC[th;VECTOR_ARITH`A- vec 0= A`;VECTOR_ARITH`(t1 % x + t2 % v + t3 % ((&1 - a) % u + a % w)) - x=(t1- &1) % x + t2 % v + t3 % ((&1 - a) % u + a % w)`])
THEN FIND_ASSUM MP_TAC `t1 + t2 + t3 = &1:real`
THEN REWRITE_TAC[REAL_ARITH`t1 + t2 + t3 = &1<=> t1- &1= (-- t2)- t3 `]
THEN DISCH_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(--t2 - t3) % x + t2 % v + t3 % ((&1 - a) % u + a % w)= t2 % (v-x) + t3 % ((&1 - a) % u + a % w-x)`]
THEN REWRITE_TAC[CROSS_LADD; CROSS_RADD; CROSS_LMUL; CROSS_RMUL;CROSS_REFL;CROSS_RNEG;CROSS_LNEG]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[GSYM CROSS_LMUL;GSYM CROSS_LADD;DOT_LMUL]
THEN MRESAL_TAC REAL_LT_LMUL_EQ [` &0:real `;`((((&1 - a) % u + a % w - x) cross (v - x)) dot (u1 - x)):real`;`t3:real`][REAL_ARITH`a * &0 = &0`]);;


let exists_cross_dot_fully_surrounded2_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real v1:real^3 u1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<a /\ a< &1
/\ fan80(x,V,E)
/\ ~collinear{x,v1,u1}
/\ v1 IN aff_gt {x} {v,(&1-a)%u+ a%w}
/\ &0< azim x v1 v u1
/\  azim x v1 v u1 < pi
==>
?t:real.  &0< t/\ t < &1
/\ (!h:real. &0<h /\ h< t 
==> &0 <
((((&1 - h) % ((&1 - a) % u + a % w) + h % u) - x) cross (v - x)) dot (u1 - x))`,

REPEAT STRIP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`(((&1 - t) % ((&1 - a) % u + a % w) + t % u) - x)=
 ((&1 - a) % u + a % w - x) + t % (--a) % ( w-u)`;]
THEN ABBREV_TAC`va=(&1 - a) % u + a % w-x:real^3`
THEN REWRITE_TAC[CROSS_LADD;CROSS_LMUL;DOT_LADD;DOT_LMUL; REAL_ARITH`&0<A+C*(--B)*D<=> (B*D)*C<A`]
THEN MRESA_TAC cross_dot_fully_surrounded2_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` v:real^3`;` u:real^3`;`w:real^3`;`a:real`;`v1:real^3`;`u1:real^3`]
THEN DISJ_CASES_TAC(REAL_ARITH`(((w - u) cross (v - x:real^3)) dot (u1- x)<= &0 )\/ (&0<((w - u) cross (v - x:real^3)) dot (u1- x))`)
THENL[ 
EXISTS_TAC`&1/ &2`
THEN REWRITE_TAC[REAL_ARITH`&0 < &1 / &2 /\ &1 / &2 < &1`]
THEN REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0 < a ==> &0<= (a:real)`) THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0 < h ==> &0<= (h:real)`) THEN RESA_TAC
THEN MRESAL_TAC REAL_LE_LMUL[`a:real`;`((w - u) cross (v - x:real^3)) dot (u1- x:real^3)`;`&0`][REAL_ARITH`(A:real)*  &0= &0`]
THEN MRESAL_TAC REAL_LE_RMUL[`a*(((w - u) cross (v - x:real^3)) dot (u1- x))`;`&0`;`h:real`][REAL_ARITH`&0 * (A:real)= &0`]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MRESA_TAC REAL_LT_MUL[`a:real`;`((w - u) cross (v - x:real^3)) dot (u1- x:real^3)`]
THEN MP_TAC(REAL_ARITH`&0<(a:real)*(((w - u) cross (v - x:real^3)) dot (u1- x))==> ~( (a:real)*(((w - u) cross (v - x:real^3)) dot (u1- x))= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_RINV`(a:real)*(((w - u) cross (v - x:real^3)) dot (u1- x))`
THEN MRESA1_TAC REAL_LT_INV`(a:real)*(((w - u) cross (v - x:real^3)) dot (u1- x))`
THEN MRESA_TAC REAL_LT_MUL[`inv((a:real)*(((w - u) cross (v - x:real^3)) dot (u1- x)))`;`((va cross (v - x)) dot (u1 - x))`]
THEN EXISTS_TAC `min ((inv(a *(((w - u) cross (v - x:real^3)) dot (u1- x)))*((va cross (v - x)) dot (u1 - x))) / &2 ) (&1/ &2)`
THEN STRIP_TAC
THENL[POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
STRIP_TAC
THENL[
 REAL_ARITH_TAC;
REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH` h< min ((inv(a *(((w - u) cross (v - x:real^3)) dot (u1- x)))*((va cross (v - x)) dot (u1 - x))) / &2 ) (&1/ &2)
==>
h< ((inv(a *(((w - u) cross (v - x:real^3)) dot (u1- x)))*((va cross (v - x)) dot (u1 - x))) / &2 )`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_LMUL[`(a *(((w - u) cross (v - x:real^3)) dot (u1- x)))`;`h:real`;`((inv(a *(((w - u) cross (v - x:real^3)) dot (u1- x)))*((va cross (v - x)) dot (u1 - x))) / &2 )`]
THEN ASM_TAC
THEN ABBREV_TAC`vb=a * (((w - u) cross (v - x:real^3)) dot (u1- x))`
THEN REPEAT DISCH_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`vb * (inv vb * ((va cross (v - x)) dot (u1 - x))) / &2=
(vb * inv vb) * ((va cross (v - x)) dot (u1 - x))/ &2`]
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC]]]);;




let properties_of_coplanar=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
~collinear{x,v,u}
/\ v1 IN aff_gt {x} {v,u}
==> coplanar{x,v1,v,u}`,
REPEAT STRIP_TAC
THEN REWRITE_TAC[coplanar]
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(u:real^3) `;]
THEN MRESA_TAC AFF_GT_1_2[`(x:real^3)` ;` (v:real^3)`;`(u:real^3) `;]
THEN POP_ASSUM(fun th-> REWRITE_TAC[th;IN_ELIM_THM])
THEN STRIP_TAC
THEN EXISTS_TAC`x:real^3` 
THEN EXISTS_TAC`v:real^3`
THEN EXISTS_TAC`u:real^3`
THEN SUBGOAL_THEN`(x:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL[REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&0`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;
SUBGOAL_THEN`(v:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL[ REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC`&0`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;
SUBGOAL_THEN`(u:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;
SUBGOAL_THEN`(v1:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3:real`
THEN ASM_REWRITE_TAC[];
ASM_TAC
THEN SET_TAC[]]]]]);;

let coplanar_is_cross_fan=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
~collinear{x,v,u}
/\ v1 IN aff_gt {x} {v,u}
==> ((v-x) cross (u-x)) dot (v1-x)= &0`,
REPEAT STRIP_TAC
THEN MRESA_TAC properties_of_coplanar[`x:real^3`;`v:real^3`;`u:real^3`;`v1:real^3`]
THEN ONCE_REWRITE_TAC[DOT_SYM;]
THEN REWRITE_TAC[DOT_CROSS_DET]
THEN ONCE_REWRITE_TAC[GSYM COPLANAR_DET_EQ_0]
THEN ASM_REWRITE_TAC[]);;

let lie_in_half_space_and_azim=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real v1:real^3 u1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<a /\ a< &1
/\ fan80(x,V,E)
/\ ~collinear{x,v1,u1}
/\ v1 IN aff_gt {x} {v,(&1-a)%u+ a%w}
/\ &0< azim x v1 v u1
/\  azim x v1 v u1 < pi
==> &0 < ((v-x) cross (u - x)) dot (v1-x)`,

REPEAT STRIP_TAC
THEN FIND_ASSUM MP_TAC `fan80(x:real^3,V,E)`
THEN REWRITE_TAC[fan80]
THEN DISCH_THEN(LABEL_TAC"yeu em")
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (u:real^3)`;`v:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] 
THEN REMOVE_THEN "yeu em" (fun th -> MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC "YEU EM"))
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - a) % (u:real^3) + a % (w:real^3)`;]
THEN MRESA_TAC AFF_GT_1_2[`(x:real^3)` ;` (v:real^3)`;`(&1 - a) % (u:real^3) + a % (w:real^3) `;]
THEN FIND_ASSUM MP_TAC `v1 IN aff_gt {x} {v,(&1-a)%u+ a%(w:real^3)}`
THEN POP_ASSUM(fun th-> REWRITE_TAC[th;IN_ELIM_THM])
THEN STRIP_TAC 
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A- vec 0= A`;VECTOR_ARITH`(t1 % x + t2 % v + t3 % ((&1 - a) % u + a % w)) - x=(t1- &1) % x + t2 % v + t3 % ((&1 - a) % u + a % w)`]
THEN FIND_ASSUM MP_TAC `t1 + t2 + t3 = &1:real`
THEN REWRITE_TAC[REAL_ARITH`t1 + t2 + t3 = &1<=> t1- &1= (-- t2)- t3 `]
THEN DISCH_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(--t2 - t3) % x + t2 % v + t3 % ((&1 - a) % u + a % w)= t2 % (v-x) + t3 % ((&1 - a) % (u-x) + a % (w-x))`]
THEN REWRITE_TAC[ DOT_RADD;DOT_RMUL;DOT_CROSS_SELF]
THEN REDUCE_ARITH_TAC
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`; `u:real^3`;`v:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[CROSS_TRIPLE]);;






let exists_cut_small_edges_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real v1:real^3 u1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<a /\ a< &1
/\ fan80(x,V,E)
/\ ~collinear{x,v1,u1}
/\ v1 IN aff_gt {x} {v,(&1-a)%u+ a%w}
/\ &0< azim x v1 v u1
/\  azim x v1 v u1 < pi
==>
?t:real.  &0< t/\ t < &1
/\ ~(aff_gt {x} {v,(&1-t)%((&1-a)%u+ a%w)+t%u} INTER aff_gt {x} {v1,u1}={})`,

REPEAT STRIP_TAC
THEN MRESA_TAC lie_in_half_space_and_azim[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`;`v1:real^3`;` u1:real^3`]
THEN MRESA_TAC exists_cross_dot_fully_surrounded1_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`;`v1:real^3`;` u1:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"MA")
THEN MRESA_TAC exists_cross_dot_fully_surrounded2_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`;`v1:real^3`;` u1:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"BE NHO")
THEN FIND_ASSUM MP_TAC `fan80(x:real^3,V,E)`
THEN REWRITE_TAC[fan80]
THEN DISCH_THEN(LABEL_TAC"yeu em")
THEN ABBREV_TAC`ta=min (t:real) (t':real)/ &2`
THEN EXISTS_TAC `(ta:real)`
THEN SUBGOAL_THEN `&0< ta:real /\ ta < &1 /\ ta< t /\ ta< t'` ASSUME_TAC
THENL(*1*)[
POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN FIND_ASSUM MP_TAC`&0<t:real`
THEN FIND_ASSUM MP_TAC`t:real < &1`
THEN FIND_ASSUM MP_TAC`&0<t':real`
THEN FIND_ASSUM MP_TAC`t':real < &1`
THEN REAL_ARITH_TAC;(*1*)

ASM_REWRITE_TAC[]
THEN MP_TAC(REAL_ARITH`ta< &1 ==> &0< (&1-ta):real`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL [`(&1-ta):real`;`a:real`]
THEN MRESA_TAC REAL_LT_LMUL [`(&1-ta):real`;`a:real`;`&1`]
THEN MP_TAC(REAL_ARITH` &0< ta /\ (&1-ta)*a< (&1-ta)* &1 :real==> (&1-ta)*a< &1`)
THEN RESA_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (u:real^3)`;`v:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] 
THEN REMOVE_THEN "yeu em" (fun th -> MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC "YEU EM"))
THEN MRESAL_TAC not_collinear_is_properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`((&1-ta)*a):real`][VECTOR_ARITH`(&1 - ((&1 - ta) * a)) % u + ((&1 - ta) * a ) % w=
(&1 - ta) % ((&1 - a) % u + a % w) + ta % u:real^3`]
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`]
THEN MRESA_TAC properties_of_collinear4_points_fan[`x:real^3`;`v:real^3`;`(&1-a)%u+ a%w:real^3`;`v1:real^3`]
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`v1:real^3`;`u1:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"CON")
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - ta) % ((&1 - a) % u + a % w) + ta % u:real^3`;]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v1:real^3)`;`u1:real^3`;]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - a) % (u:real^3) + a % (w:real^3)`;]
THEN MRESA_TAC AFF_GT_1_2[`(x:real^3)` ;` (v1:real^3)`;`(u1:real^3) `;]
THEN MRESA_TAC AFF_GT_1_2[`(x:real^3)` ;` (v:real^3)`;`(&1 - ta) % ((&1 - a) % u + a % w) + ta % u:real^3 `;]
THEN ASM_REWRITE_TAC[INTER;IN_ELIM_THM;EXTENSION;EMPTY;IN;NOT_FORALL_THM]
THEN ABBREV_TAC`a1=(v-x):real^3`
THEN ABBREV_TAC`a2=(((&1-ta)%((&1-a)%u+ a%w)+ta%u)-x):real^3`
THEN ABBREV_TAC`a3=(v1-x) :real^3`
THEN ABBREV_TAC`a4=u1-x:real^3`
THEN ABBREV_TAC`va=a1 cross a2:real^3`
THEN ABBREV_TAC`vb=a3 cross a4:real^3`
THEN EXISTS_TAC`(vb:real^3) cross (va:real^3)+(x:real^3)`
THEN STRIP_TAC
THENL(*2*)[
EXISTS_TAC`&1-(vb:real^3) dot (a2:real^3)+ vb dot (a1:real^3)`
THEN EXISTS_TAC`(vb:real^3) dot (a2:real^3)`
THEN EXISTS_TAC`--((vb:real^3) dot (a1:real^3))`
THEN REMOVE_THEN "MA"(fun th->  MRESA1_TAC th `ta:real`)
THEN ASM_REWRITE_TAC[REAL_ARITH`(&1 - vb dot a2 + vb dot a1) + vb dot a2 + --(vb dot a1) = &1`]
THEN SUBGOAL_THEN `&0< --((vb:real^3) dot (a1:real^3))` ASSUME_TAC
THENL(*3*)[EXPAND_TAC"vb"
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[DOT_LNEG]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN REMOVE_THEN "CON"MP_TAC
THEN REAL_ARITH_TAC;(*3*)

ASM_REWRITE_TAC[]
THEN EXPAND_TAC"va"
THEN REWRITE_TAC[CROSS_LAGRANGE;VECTOR_MUL_LNEG]
THEN EXPAND_TAC"a1"
THEN EXPAND_TAC"a2"
THEN VECTOR_ARITH_TAC];(*2*)

ONCE_REWRITE_TAC[CROSS_SKEW]
THEN EXPAND_TAC"vb"
THEN REWRITE_TAC[CROSS_LAGRANGE;]
THEN EXISTS_TAC`&1+(va:real^3) dot (a4:real^3)- va dot (a3:real^3)`
THEN EXISTS_TAC`--(va:real^3) dot (a4:real^3)`
THEN EXISTS_TAC`((va:real^3) dot (a3:real^3))`
THEN REMOVE_THEN "BE NHO"(fun th->  MRESA1_TAC th `ta:real`)
THEN ASM_REWRITE_TAC[DOT_LNEG;VECTOR_MUL_LNEG;REAL_ARITH`(&1 + va dot a4 - va dot a3) + --(va dot a4) + va dot a3 = &1`;]

THEN STRIP_TAC
THENL(*3*)[
EXPAND_TAC"va"
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[DOT_LNEG]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;(*3*)

STRIP_TAC 
THENL(*4*)[
EXPAND_TAC"va"
THEN EXPAND_TAC"a2"
THEN REWRITE_TAC[VECTOR_ARITH`((&1 - ta) % ((&1 - a) % u + a % w) + ta % u) - x
=(&1 - ta) % ((&1 - a) % u + a % w-x) + ta % (u - x)`]
THEN ABBREV_TAC`vu=(&1 - a) % u + a % w - x:real^3`
THEN REWRITE_TAC[CROSS_RADD;CROSS_RMUL;DOT_LADD;DOT_LMUL;]
THEN MRESAL_TAC coplanar_is_cross_fan[`x:real^3`;`v:real^3`;`(&1 - a) % u + a % w:real^3`;`v1:real^3`]
[VECTOR_ARITH`((&1 - a) % u + a % w) - x=(&1 - a) % u + a % w - x`]
THEN REDUCE_ARITH_TAC
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[];(*4*)
EXPAND_TAC"a3"
THEN EXPAND_TAC"a4"
THEN REWRITE_TAC[VECTOR_ARITH`-- A+B= B-A:real^3`;VECTOR_ARITH`(&1+A-B)%X+B%U-A%V=X-A%(V-X)+B%(U-X)`]
THEN VECTOR_ARITH_TAC]]]]);;

let aff_gt_subset_aff_ge=prove(`!x:real^3 v:real^3 u:real^3.
DISJOINT {x} {v,u}
==>
aff_gt {x} {v,u} SUBSET aff_ge {x} {v,u}`,
REPEAT STRIP_TAC
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM;SUBSET]
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3:real`
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REAL_ARITH_TAC);;


let aff_gt1_subset_aff_ge=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
DISJOINT {x} {v,u} /\ ~collinear {x,v1,u}
/\ v1 IN aff_ge {x} {v,u}
==>
aff_gt {x} {v1,u} SUBSET aff_ge {x} {v,u}`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v1:real^3`;`u:real^3`]
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v1:real^3`;`u:real^3`][IN_ELIM_THM;SUBSET]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % (t1 % x + t2 % v + t3 % u) + t3' % u
=(t1'+ t2'*t1) % x + (t2'* t2) % v + (t2' * t3  + t3') % u:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`t1' + t2' * t1:real`
THEN EXISTS_TAC`t2' * t2:real`
THEN EXISTS_TAC`t2' * t3 +t3':real`
THEN MP_TAC(REAL_ARITH`&0<t2'==> &0<= t2'`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t2:real`]
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t3:real`]
THEN ASM_REWRITE_TAC[REAL_ARITH`(t1' + t2' * t1) + t2' * t2 + t2' * t3 + t3'=
t1'+t2'*(t1+t2+t3)+t3'`; REAL_ARITH`A* &1=A`]
THEN ASM_TAC
THEN REAL_ARITH_TAC);;

let aff_gt12_subset_aff_ge=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
DISJOINT {x} {v,u} /\ ~collinear {x,v1,v}
/\ v1 IN aff_ge {x} {v,u}
==>
aff_gt {x} {v1,v} SUBSET aff_ge {x} {v,u}`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v1:real^3`;`v:real^3`]
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v1:real^3`;`v:real^3`][IN_ELIM_THM;SUBSET]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % (t1 % x + t2 % v + t3 % u) + t3' % v
=(t1'+ t2'*t1) % x + (t2'* t2+t3') % v + (t2' * t3 ) % u:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`t1' + t2' * t1:real`
THEN EXISTS_TAC`t2' * t2 +t3':real`
THEN EXISTS_TAC`t2' * t3 :real`
THEN MP_TAC(REAL_ARITH`&0<t2'==> &0<= t2'`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t2:real`]
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t3:real`]
THEN ASM_REWRITE_TAC[REAL_ARITH`(t1' + t2' * t1) + (t2' * t2 +t3')+ t2' * t3 =
t1'+t2'*(t1+t2+t3)+t3'`; REAL_ARITH`A* &1=A`]
THEN ASM_TAC
THEN REAL_ARITH_TAC);;

let aff_gt2_subset_aff_ge=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
DISJOINT {x} {v,u} /\ ~collinear {x,v1,u} /\ ~collinear {x,v1,v}
/\ v1 IN aff_gt {x} {v,u}
==>
azim x v1 v u= pi`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v1:real^3`;`u:real^3`]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESA_TAC 	AZIM_EQ_PI[`x:real^3`;`v1:real^3`;`v:real^3`;`u:real^3`]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`v1 = t1 % x + t2 % v + t3 % u
<=> t2 % v = (--t1) % x + v1 + (--t3) % u`]
THEN MP_TAC(REAL_ARITH`&0<t2==> ~( t2= &0)`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0<t3==> -- t3< &0`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`t2:real`
THEN MRESA1_TAC REAL_LT_INV`t2:real`
THEN MRESAL_TAC REAL_LT_LMUL[`inv t2:real`;`-- t3:real`;`&0`][REAL_ARITH`A * &0= &0`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`t2 % v = --t1 % x + v1 + --t3 % u:real^3 ==> (inv (t2))%(t2 % v ) = (inv (t2))%( --t1 % x + v1 + --t3 % u)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C+D)=A%B+A%C+A%D`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN MRESAL_TAC  AFF_LT_2_1[`x:real^3`;`v1:real^3`;`u:real^3`][IN_ELIM_THM;SUBSET]
THEN EXISTS_TAC`(inv t2 * --t1):real`
THEN EXISTS_TAC`inv t2:real`
THEN EXISTS_TAC`(inv t2 * --t3):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv t2 * (--t1) + inv t2 + inv t2 * (--t3)=
inv t2 * (t2+ &1 -(t1 +t2 +t3))`; REAL_ARITH`A+ &1- &1=A`]);;

let remove_variable_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 t1:real t2:real t3:real.
 &0 < t3 
/\ w = t1 % x + t2 % v + t3 % u
==> u= inv(t3) % w - (inv(t3)*t1) % x- (inv(t3) * t2) % v`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[VECTOR_ARITH`w = t1'' % x + t2'' % v + t3'' % u <=> t3'' % u = w-t1'' % x - t2'' % (v:real^3)`]
THEN MP_TAC(REAL_ARITH `&0 < (t3:real) ==>  ~(t3 = &0)`)
THEN MP_TAC(ISPEC`(t3:real)`REAL_LT_INV)
THEN POP_ASSUM(fun th-> REWRITE_TAC[th] THEN ASSUME_TAC(th))
THEN STRIP_TAC THEN STRIP_TAC 
THEN MP_TAC(ISPEC`(t3:real)`REAL_MUL_LINV)
THEN POP_ASSUM(fun th-> REWRITE_TAC[th] THEN ASSUME_TAC(th))
THEN STRIP_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t3 % u = w-t1 % x - t2 % v:real^3
==> (inv (t3))%(t3 % u) = (inv (t3))%( w-t1 % x - t2 % v:real^3)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[th;VECTOR_ARITH`A%B%C= (A*B)%C`;VECTOR_ARITH`&1 %A=A`;VECTOR_ARITH`A%(B-C-D)=A%B-A%C-A%D`] THEN ASSUME_TAC(th)));;


let aff_gt_inter_aff_gt=prove(`!(x:real^3) (v:real^3) (w:real^3).
~collinear {x,v,w}
==>
aff_gt {x} {v , w} = aff_gt {x , v} {w} INTER aff_gt {x , w} {v}`,

 REPEAT STRIP_TAC THEN MRESA_TAC th3 [`x:real^3`;`v:real^3`;`w:real^3`]
   THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`]AFF_GT_1_2)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
		THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`]AFF_GT_2_1)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
		THEN MP_TAC(ISPECL[`x:real^3`;`w:real^3`;`v:real^3`]AFF_GT_2_1)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
   THEN ASM_REWRITE_TAC[INTER;IN_ELIM_THM;EXTENSION]THEN GEN_TAC THEN EQ_TAC 
THENL(*1*)[
STRIP_TAC THEN STRIP_TAC
THENL(*2*)[
    EXISTS_TAC `t1:real` THEN EXISTS_TAC `t2:real` THEN EXISTS_TAC `t3:real` THEN ASM_MESON_TAC[];
EXISTS_TAC `(t1:real)` THEN
         EXISTS_TAC `(t3:real)` THEN EXISTS_TAC `(t2:real)` 
THEN
         ASM_MESON_TAC[REAL_ARITH `(t1:real)+ (t3:real) +(t2:real)=t1 + t2 + t3`;VECTOR_ARITH ` t1 % x + t2 % v + t3 % w = (t1:real) % (x:real^3) + (t3:real) % (w:real^3) + (t2:real) % (v:real^3)`]](*2*);

         STRIP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC 
	   THEN POP_ASSUM(fun th-> GEN_REWRITE_TAC(PATH_CONV "rrlr" o ONCE_DEPTH_CONV )[th] THEN ASSUME_TAC(th)) THEN POP_ASSUM MP_TAC 
	   THEN POP_ASSUM(fun th-> GEN_REWRITE_TAC(PATH_CONV "rrlr" o ONCE_DEPTH_CONV )[SYM(th)] THEN ASSUME_TAC(th))
THEN DISJ_CASES_TAC(SET_RULE`t3 - t2' = &0 \/ ~((t3:real) - (t2':real) = &0) `)
THENL[POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`A-B= &0 <=> A=B`]
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`t1':real`
THEN EXISTS_TAC`t3':real`
THEN EXISTS_TAC`t2':real`
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % w + t3' % v = t1' % x + t3' % v + t2' % w`;
REAL_ARITH`t1' + t3' + t2'=t1' + t2' + t3'`]
THEN ASM_TAC THEN REAL_ARITH_TAC;

REWRITE_TAC[VECTOR_ARITH
 `a % x + b % y + c % z= a1 % x + b1 % z + c1 % y <=> (c-b1) % z = (a1-a) % x + (c1-b)% y`]	   
	   THEN REWRITE_TAC[REAL_ARITH`a+b+c=a1+b1+c1<=> c1-b=(a-a1)+(c-b1)`]
THEN MRESA1_TAC REAL_MUL_LINV`t3 - t2'`
	   THEN DISCH_TAC THEN DISCH_TAC 	   THEN DISCH_TAC THEN DISCH_TAC 
THEN MP_TAC(SET_RULE`
(t3 - t2') % w = (t1' - t1) % x + (t3' - t2) %  v:real^3
==> (inv (t3 - t2'))%((t3 - t2') % w ) = (inv (t3 - t2'))%((t1' - t1) % x + (t3' - t2) %  v:real^3)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C= (A*B)%C`;VECTOR_ARITH`&1 %A=A`;VECTOR_ARITH`A%(B+C)=A%B+A%C`] THEN ASSUME_TAC(SYM(th)))
THEN STRIP_TAC
THEN SUBGOAL_THEN`w IN aff{(x:real^3),v}` ASSUME_TAC
THENL[REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM;] THEN EXISTS_TAC`inv(t3-t2') *(t1'-t1)`
THEN EXISTS_TAC`inv(t3-t2') *(t3'-t2)` THEN ASM_REWRITE_TAC[REAL_ARITH`A*B+A*C=A*(B+C)`];

ASM_SET_TAC[]]]]);;


let aff_gt3_subset_aff_gt=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
DISJOINT {x} {v,u} /\ ~collinear{x,v,v1}
/\ v1 IN aff_gt {x} {v,u}
==>
aff_gt {x} {v,v1} SUBSET aff_gt {x} {v,u}`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`v1:real^3`]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`v1:real^3`][IN_ELIM_THM;SUBSET]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % v + t3' % (t1 % x + t2 % v + t3 % u)
=(t1'+ t3'*t1) % x + (t2'+ t3' * t2) % v + (t3' * t3) % u:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`t1' + t3' * t1:real`
THEN EXISTS_TAC`t2' + t3' * t2:real`
THEN EXISTS_TAC`t3' * t3:real`
THEN MRESA_TAC REAL_LT_MUL[`t3':real`;`t2:real`]
THEN MRESA_TAC REAL_LT_MUL[`t3':real`;`t3:real`]
THEN ASM_REWRITE_TAC[REAL_ARITH`(t1' + t3' * t1) + (t2' + t3' * t2) + t3' * t3=
t1'+ t2' + t3'*(t1+t2+t3)`; REAL_ARITH`A* &1=A`]
THEN ASM_TAC
THEN REAL_ARITH_TAC);;



let aff_ge1_subset_aff_ge=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
DISJOINT {x} {v,u} /\ ~collinear{x,v1,u}
/\ v1 IN aff_ge {x} {v,u}
==>
aff_ge {x} {v1,u} SUBSET aff_ge {x} {v,u}`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v1:real^3`;`u:real^3`]
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v1:real^3`;`u:real^3`][IN_ELIM_THM;SUBSET]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % (t1 % x + t2 % v + t3 % u) + t3' % u
=(t1'+ t2'*t1) % x + (t2'* t2) % v + (t2' * t3  + t3') % u:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`t1' + t2' * t1:real`
THEN EXISTS_TAC`t2' * t2:real`
THEN EXISTS_TAC`t2' * t3 +t3':real`
THEN MP_TAC(REAL_ARITH`&0<t2'==> &0<= t2'`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t2:real`]
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t3:real`]
THEN ASM_REWRITE_TAC[REAL_ARITH`(t1' + t2' * t1) + t2' * t2 + t2' * t3 + t3'=
t1'+t2'*(t1+t2+t3)+t3'`; REAL_ARITH`A* &1=A`]
THEN ASM_TAC
THEN REAL_ARITH_TAC);;

let aff_ge_1_1_subset_aff_ge_fan=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
DISJOINT {x} {v,u} /\ ~(x=v1)
/\ v1 IN aff_ge {x} {v,u}
==>
aff_ge {x} {v1} SUBSET aff_ge {x} {v,u}`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GE_1_1[`x:real^3`;`v1:real^3`][IN_ELIM_THM;SUBSET]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % (t1 % x + t2 % v + t3 % u)
=(t1'+ t2'*t1) % x + (t2'* t2) % v + (t2' * t3 ) % u:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`t1' + t2' * t1:real`
THEN EXISTS_TAC`t2' * t2:real`
THEN EXISTS_TAC`t2' * t3 :real`
THEN MP_TAC(REAL_ARITH`&0<t2'==> &0<= t2'`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t2:real`]
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t3:real`]
THEN ASM_REWRITE_TAC[REAL_ARITH`(t1' + t2' * t1) + t2' * t2 + t2' * t3 =
t1'+t2'*(t1+t2+t3)`; REAL_ARITH`A* &1=A`]);;


let decomposition_planar_by_angle_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3.
~ collinear {x,v,u} /\ ~collinear {x,v,w}
/\ w IN aff_ge {x,v} {u}
==> u IN aff_gt {x} {v,w} \/ w IN aff_ge {x} {v,u}`,
REPEAT STRIP_TAC
THEN MRESAL_TAC aff_ge_inter_aff_ge[`(x:real^3)`;`(v:real^3)`;`(u:real^3)`][INTER; IN_ELIM_THM]
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`v:real^3`;`w:real^3`; `u:real^3`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th)])
THEN MRESA_TAC AZIM_EQ_0_ALT[`x:real^3`;`v:real^3`;`w:real^3`; `u:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`w:real^3`]
THEN MRESAL_TAC  AFF_GT_2_1[`x:real^3`;`v:real^3`;`w:real^3`][IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(&0 < t2) \/   &0 <= --(t2:real)`)
THENL[
SUBGOAL_THEN `u IN aff_gt {x} {v,w:real^3}` ASSUME_TAC
THENL[
MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`w:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3:real`
THEN MP_TAC(REAL_ARITH`&0< t3==> &0 <= t3:real`)
THEN ASM_REWRITE_TAC[];
POP_ASSUM MP_TAC THEN SET_TAC[]];
MRESA_TAC remove_variable_fan[`x:real^3`; `v:real^3`; `w:real^3`;`u:real^3`;`t1:real`;`t2:real`;`t3:real`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th);])
THEN DISCH_TAC
THEN REWRITE_TAC[VECTOR_ARITH`inv t3 % u - (inv t3 * t1) % x - (inv t3 * t2) % v
=(--inv t3 * t1) % x + inv t3 % u + (inv t3 * (--t2)) % v`]
THEN MP_TAC(REAL_ARITH`&0<t3==> ~( t3= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`t3:real`
THEN MRESA1_TAC REAL_LT_INV`t3:real`
THEN MP_TAC(REAL_ARITH`&0< inv t3==> &0 <= inv t3`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LE_MUL[`inv t3:real`;`-- (t2:real)`]
THEN DISCH_TAC
THEN SUBGOAL_THEN `w IN aff_ge {x, u} {v:real^3}` ASSUME_TAC
THENL[ 
MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESAL_TAC  AFF_GE_2_1[`x:real^3`;`u:real^3`;`v:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`(--inv t3 * t1):real`
THEN EXISTS_TAC`inv t3:real`
THEN EXISTS_TAC`(inv t3 * --t2):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`--inv t3 * t1 + inv t3 + inv t3 * --t2=
inv t3 * (t3+ &1- (t1 +t2 + t3))`; REAL_ARITH`a + &1 - &1 =a`];
POP_ASSUM MP_TAC THEN SET_TAC[]]]);;






let properties_of_fan7=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 v1:real^3  u1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {v1,u1} IN E /\ v IN aff_ge {x} {v1,u1}
==> v = v1 \/ v = u1`,

REPEAT STRIP_TAC
THEN FIND_ASSUM MP_TAC `FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN; fan6; fan7]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` u:real^3`;`v:real^3`]
THEN STRIP_TAC 
  THEN POP_ASSUM  (fun th-> MP_TAC(ISPECL[`{(v:real^3),(u:real^3)}`;`{(v1:real^3),(u1:real^3)}`]th)) 
  THEN ASM_REWRITE_TAC[UNION;IN_ELIM_THM;]
THEN SUBGOAL_THEN `(v:real^3) IN aff_ge {(x:real^3)} {(v:real^3),(u:real^3)}` ASSUME_TAC
THENL(*1*)[
 MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (u:real^3)`]AFF_GE_1_2) 
   THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC `&0:real` 
  THEN EXISTS_TAC `&1:real` THEN EXISTS_TAC `&0:real`  
THEN REWRITE_TAC[VECTOR_ARITH`w= &0 % x+ &1 % w + &0 % v `] THEN REAL_ARITH_TAC;

DISCH_TAC
THEN MP_TAC(SET_RULE`v IN aff_ge {x} {v, u} /\ v IN aff_ge {x} {v1, u1} ==> v IN aff_ge {x} {v, u} INTER aff_ge {x:real^3} {v1, u1:real^3}`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`{v, u} INTER {v1, u1:real^3}={} \/ ~({v, u} INTER {v1, u1}={})`)
THENL(*2*)[
ASM_REWRITE_TAC[]
THEN SUBGOAL_THEN`aff_ge {x:real^3} {}= {y| ?t1. t1= &1 /\ y= t1 %x}` ASSUME_TAC
THENL(*3*)[
ASSUME_TAC(SET_RULE `DISJOINT {x:real^3} {}`)
THEN AFF_TAC;

ASM_REWRITE_TAC[IN_ELIM_THM]
THEN STRIP_TAC THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[] THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]];(*2*)

DISJ_CASES_TAC(SET_RULE`~(v IN {v, u} INTER {v1, u1:real^3}) \/ (v IN {v, u} INTER {v1, u1})`)
THENL(*3*)[
SUBGOAL_THEN`({v:real^3, u} INTER {v1, u1}={u})` ASSUME_TAC
THENL(*4*)[
 ASM_TAC
THEN SET_TAC[];(*4*)

ASM_REWRITE_TAC[]
THEN SUBGOAL_THEN` aff_ge {(x:real^3)} {(u:real^3)} SUBSET aff {x,u}` ASSUME_TAC
THENL(*5*)[
MP_TAC(ISPECL[`(x:real^3)`;` (u:real^3)`;]AFF_GE_1_1) 
THEN MP_TAC(SET_RULE`~((x:real^3) = (u:real^3))==> DISJOINT {x} {u}`)
  THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
  THEN ASM_REWRITE_TAC[]
  THEN DISCH_TAC THEN ASM_REWRITE_TAC[aff;AFFINE_HULL_2] THEN ASM_TAC THEN SET_TAC[];(*5*)

FIND_ASSUM MP_TAC`{v,u:real^3} IN E`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN DISCH_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` v:real^3`;`u:real^3`]
THEN ASM_TAC
THEN SET_TAC[]]];(*3*)

POP_ASSUM MP_TAC
THEN SET_TAC[]]]])  ;;





let properties1_of_fan7=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 v1:real^3  u1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {v1,u1} IN E /\ v IN aff_ge {x} {v1}
==> v = v1`,

REPEAT STRIP_TAC
THEN FIND_ASSUM MP_TAC `FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN; fan6; fan7]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` u:real^3`;`v:real^3`]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` u1:real^3`;`v1:real^3`]
THEN MP_TAC(SET_RULE`(v1:real^3) IN V==> (?v. v IN V /\ {v1} = {v})`)
THEN REDA_TAC
THEN STRIP_TAC 
  THEN POP_ASSUM  (fun th-> MP_TAC(ISPECL[`{(v:real^3),(u:real^3)}`;`{(v1:real^3)}`]th)) 
  THEN ASM_REWRITE_TAC[UNION;IN_ELIM_THM;]
THEN SUBGOAL_THEN `(v:real^3) IN aff_ge {(x:real^3)} {(v:real^3),(u:real^3)}` ASSUME_TAC
THENL(*1*)[
 MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (u:real^3)`]AFF_GE_1_2) 
   THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC `&0:real` 
  THEN EXISTS_TAC `&1:real` THEN EXISTS_TAC `&0:real`  
THEN REWRITE_TAC[VECTOR_ARITH`w= &0 % x+ &1 % w + &0 % v `] THEN REAL_ARITH_TAC;
DISCH_TAC
THEN MP_TAC(SET_RULE`v IN aff_ge {x} {v, u:real^3} /\ v IN aff_ge {x} {v1} ==> v IN aff_ge {x} {v, u} INTER aff_ge {x:real^3} {v1}`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`{v, u:real^3} INTER {v1}={} \/ ~({v, u} INTER {v1}={})`)
THENL(*2*)[
ASM_REWRITE_TAC[]
THEN SUBGOAL_THEN`aff_ge {x:real^3} {}= {y| ?t1. t1= &1 /\ y= t1 %x}` ASSUME_TAC
THENL(*3*)[
ASSUME_TAC(SET_RULE `DISJOINT {x:real^3} {}`)
THEN AFF_TAC;
ASM_REWRITE_TAC[IN_ELIM_THM]
THEN STRIP_TAC THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[] THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]];(*2*)

DISJ_CASES_TAC(SET_RULE`~(v IN {v, u} INTER {v1:real^3}) \/ (v IN {v, u} INTER {v1})`)
THENL(*3*)[
SUBGOAL_THEN`({v:real^3, u} INTER {v1:real^3}={u})` ASSUME_TAC
THENL(*4*)[
 ASM_TAC
THEN SET_TAC[];
ASM_REWRITE_TAC[]
THEN SUBGOAL_THEN` aff_ge {(x:real^3)} {(u:real^3)} SUBSET aff {x,u}` ASSUME_TAC
THENL(*5*)[
MP_TAC(ISPECL[`(x:real^3)`;` (u:real^3)`;]AFF_GE_1_1) 
THEN MP_TAC(SET_RULE`~((x:real^3) = (u:real^3))==> DISJOINT {x} {u}`)
  THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
  THEN ASM_REWRITE_TAC[]
  THEN DISCH_TAC THEN ASM_REWRITE_TAC[aff;AFFINE_HULL_2] THEN ASM_TAC THEN SET_TAC[];
FIND_ASSUM MP_TAC`{v,u:real^3} IN E`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN DISCH_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`]
THEN ASM_TAC
THEN SET_TAC[]]];(*3*)

POP_ASSUM MP_TAC
THEN SET_TAC[]]]])  ;;


let point_in_aff_ge=prove(`!(x:real^3) (v:real^3) (w:real^3).
~collinear {x,v,w}
==>
x IN aff_ge {x} {v,w}
/\ v IN aff_ge {x} {v,w}
/\ w IN aff_ge {x} {v,w}`,

REPEAT GEN_TAC THEN STRIP_TAC THEN MRESA_TAC th3 [`x:real^3`;`v:real^3`;`w:real^3`]
  THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`]AFF_GE_1_2)
THEN RESA_TAC
		THEN ASM_REWRITE_TAC[IN_ELIM_THM]
THEN STRIP_TAC
THENL[ EXISTS_TAC`&1:real`
THEN EXISTS_TAC`&0:real`
THEN EXISTS_TAC`&0:real`
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC;  
STRIP_TAC
THENL[ EXISTS_TAC`&0:real`
THEN EXISTS_TAC`&1:real`
THEN EXISTS_TAC`&0:real`
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC;

EXISTS_TAC`&0:real`
THEN EXISTS_TAC`&0:real`
THEN EXISTS_TAC`&1:real`
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC]]);;  


let aff_ge_subset_aff_gt_union_aff_ge=prove(`!(x:real^3) (v:real^3) (w:real^3).
~collinear {x,v,w}
==>
aff_ge {x} {v , w} SUBSET  (aff_gt {x , v} {w}) UNION (aff_ge {x} {v})`,
REPEAT STRIP_TAC THEN MRESA_TAC th3 [`x:real^3`;`v:real^3`;`w:real^3`]
  THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`]AFF_GE_1_2)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
		THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`]AFF_GT_2_1)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
		THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`]AFF_GE_1_1)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[SUBSET; UNION;IN_ELIM_THM]
  THEN GEN_TAC THEN
REWRITE_TAC[REAL_ARITH `(&0 <= (t3:real)) <=> (&0 < t3) \/ ( t3 = &0)`; TAUT `(a \/ b) /\ (c \/ d) /\ e /\ f <=> ((a \/ b)/\ c /\ e /\ f) \/ ((a \/ b) /\ d /\ e /\ f)`; EXISTS_OR_THM] THEN
MATCH_MP_TAC MONO_OR THEN
SUBGOAL_THEN `((?t1:real t2:real t3:real.
       (&0 < t2 \/ t2 = &0) /\
        &0< t3 /\
       t1 + t2 + t3 = &1 /\
       (x':real^3) = t1 % x + t2 % v + t3 % w)
  ==> (?t1 t2 t3.
            &0< t3 /\ t1 + t2 + t3 = &1 /\ x' = t1 % x + t2 % v + t3 % w))` ASSUME_TAC
THENL  
[MESON_TAC[];
     ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN
     REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN GEN_TAC THEN
     REWRITE_TAC[REAL_ARITH `(&0< (t2:real) \/ (t2 = &0)) <=> ( &0<= t2)`] THEN STRIP_TAC THEN POP_ASSUM MP_TAC THEN
     POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[] THEN
     REWRITE_TAC [REAL_ARITH `(a:real)+ &0 = a`; VECTOR_ARITH  `&0 % (w:real^3) = vec 0`; 
     VECTOR_ARITH `  ((x':real^3) = (t1:real) % (x:real^3) + (t2:real) % (v:real^3) + vec 0)<=> ( x' = t1 % x + t2 % v )` ]
       THEN MESON_TAC[]]);;	

let pos_in_aff_ge_fan=prove(`!x:real^3 v:real^3 u:real^3 a:real.
DISJOINT {x} {v,u}
/\ &0<a /\ a< &1
==>
(&1-a)%v + a % u IN aff_ge {x} {v,u:real^3}`,

REPEAT STRIP_TAC
THEN MRESAL_TAC AFF_GE_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1 -a:real`
THEN EXISTS_TAC`a:real`
THEN MP_TAC(REAL_ARITH`&0< a /\ a  < &1 ==> &0 <= &1 - a /\ &0 <= a`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC);;


let aff_gt1_subset_aff_gt=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
DISJOINT {x} {v,u} /\ ~collinear {x,v1,u}
/\ v1 IN aff_gt {x} {v,u}
==>
aff_gt {x} {v1,u} SUBSET aff_gt {x} {v,u}`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v1:real^3`;`u:real^3`]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v1:real^3`;`u:real^3`][IN_ELIM_THM;SUBSET]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % (t1 % x + t2 % v + t3 % u) + t3' % u
=(t1'+ t2'*t1) % x + (t2'* t2) % v + (t2' * t3  + t3') % u:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`t1' + t2' * t1:real`
THEN EXISTS_TAC`t2' * t2:real`
THEN EXISTS_TAC`t2' * t3 +t3':real`
THEN MP_TAC(REAL_ARITH`&0<t2'==> &0<= t2'`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`t2':real`;`t2:real`]
THEN MRESA_TAC REAL_LT_MUL[`t2':real`;`t3:real`]
THEN ASM_REWRITE_TAC[REAL_ARITH`(t1' + t2' * t1) + t2' * t2 + t2' * t3 + t3'=
t1'+t2'*(t1+t2+t3)+t3'`; REAL_ARITH`A* &1=A`]
THEN ASM_TAC
THEN REAL_ARITH_TAC);;




let not_cut_inside_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<a /\ a< &1
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)

/\ (!h:real. &0<h /\ h< a
==> aff_gt{x} {v,(&1-h)%u+ h % w} INTER {v | ?e. e IN E /\ v IN aff_ge {x} e}={})
==> aff_gt{x} {v,(&1-a)%u+ a % w} INTER {v | ?e. e IN E /\ v IN aff_ge {x} e}={}`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"EM")
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> MP_TAC (th) THEN ASSUME_TAC(th))
THEN REWRITE_TAC[fan80]
THEN DISCH_THEN(LABEL_TAC"BE")
THEN DISCH_THEN(LABEL_TAC"YEU")
THEN REWRITE_TAC[INTER;EXTENSION;IN_ELIM_THM]
THEN GEN_TAC
THEN EQ_TAC
THENL(*1*)[
STRIP_TAC 
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC expand_edge_graph_fan [`(x:real^3)`;` (V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(e:real^3->bool)`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"A")
THEN DISCH_TAC
THEN REMOVE_THEN "A" MP_TAC
THEN POP_ASSUM(fun th->REWRITE_TAC[(th)])
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"A")
THEN DISCH_TAC
THEN REMOVE_THEN "A" MP_TAC
THEN REMOVE_THEN "BE" (fun th->  MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC"BE") )
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - a) % (u:real^3) + a % (w:real^3)`;]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w':real^3)`;
` (v':real^3)`]  
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v':real^3`;`w':real^3`][IN_ELIM_THM]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "VUT1")
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`(&1-a)%u +a%w:real^3`][IN_ELIM_THM]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "VUT")
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "VUT1")
THEN DISCH_TAC THEN DISCH_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(t3'= &0) \/ ~(t3':real= &0)`)
THENL(*2*)[
ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN REDUCE_VECTOR_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(t2'= &0) \/ ~(t2':real= &0)`)
THENL(*3*)[
ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN REDUCE_VECTOR_TAC
THEN DISCH_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(REAL_ARITH`&0<t3:real==> ~(t3= &0)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(t3:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN FIND_ASSUM (fun th-> REWRITE_TAC[SYM(th)])`t1 + t2 + t3 = &1:real`
THEN REWRITE_TAC[VECTOR_ARITH`t1 % x + t2 % v + t3 % (((t1 + t2 + t3) - a) % u + a % w) =
 (t1 + t2 + t3) % x<=>(t3) % (((t1 + t2 + t3) - a) % u + a % w- x)=(-- t2) % (v-x) `]
THEN ASM_REWRITE_TAC[IN;EMPTY]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
(t3) % ((&1 - a) % u + a % w- x)=(-- t2) % (v-x)
==> (inv (t3))%((t3) % ((&1 - a) % u + a % w- x) ) = (inv (t3))%((-- t2) % (v-x)):real^3`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`;]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[VECTOR_ARITH`E+A-B=C%(D-B)<=>E+A=(&1-C)%B+C%D`;]
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC `~((&1 - a) % u + a % w IN aff {x, v:real^3})`
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN EXISTS_TAC`&1-(inv t3 * (--t2)):real`
THEN EXISTS_TAC`(inv t3 * (--t2)):real`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC;(*3*)

FIND_ASSUM MP_TAC`t1 + t2 + t3 = &1:real`
THEN REWRITE_TAC[REAL_ARITH`t1 + t2 + t3 = &1<=> t1= &1 -t2 -t3`; REAL_ARITH`A+B= &1<=> A= &1-B`]
THEN DISCH_THEN(LABEL_TAC"MAI")
THEN DISCH_THEN (LABEL_TAC"YEU EM")
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(&1 - t2 - t3) % x + t2 % v + t3 % ((&1 - a) % u + a % w) =
 (&1 - t2') % x + t2' % v'<=> t2' % v'=(t2' - t2 - t3) % x + t2 % v + t3 % ((&1 - a) % u + a % w):real^3 `]
THEN MP_TAC(ISPEC`(t2':real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2' % v'=(t2' - t2 - t3) % x + t2 % v + t3 % ((&1 - a) % u + a % w):real^3
==> (inv (t2'))%(t2' % v' ) = (inv (t2'))%((t2' - t2 - t3) % x + t2 % v + t3 % ((&1 - a) % u + a % w):real^3)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C+D)=(A%B+A%C+A%D)`;REAL_ARITH`a*(b-c-d)=a*b-a*c-a*d`]
THEN REDUCE_VECTOR_TAC
THEN DISCH_TAC
THEN SUBGOAL_THEN `~coplanar{x:real^3,v':real^3,v:real^3,u:real^3}` ASSUME_TAC
THENL(*4*)[

POP_ASSUM MP_TAC
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3, v', v, u}= {(--x) +x, (--x) +v', (--x) +v, (--x) +u}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3,v':real^3,v:real^3,u:real^3}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)] THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,C,B}`])
THEN DISCH_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th;VECTOR_ARITH`((&1 - inv t2' * t2 - inv t2' * t3) % x +
                         (inv t2' * t2) % v +
                         (inv t2' * t3) % ((&1 - a) % u + a % w)) -
                        x=(inv t2' * t2) % (v-x) +
                         (inv t2' * t3) % ((&1 - a) % u + a % w-x)`])
THEN MP_TAC(REAL_ARITH`&0< t2 /\ &0< t3 /\ t1+t2+t3= &1==>  &0 < &1- t1 /\ ~(&1- t1 = &0)/\ t2+t3= &1- t1`)
THEN REMOVE_THEN"MAI" MP_TAC
THEN DISCH_TAC
THEN REMOVE_ASSUM_TAC
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ISPEC`&1- (t1:real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`&1- (t1:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2 + t3 = &1 - t1:real
==> (inv ( &1 - t1))*(t2 + t3) = (inv ( &1 - t1))*( &1 - t1:real)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C)= &1<=> A*B = &1 -A*C`]
THEN STRIP_TAC
THEN MP_TAC(REAL_ARITH` ~(t2' = &0) /\ &0 <= t2'==> &0 <t2'`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`  &0 < a==> ~(a = &0) `)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`t2':real`;`inv (&1- t1):real`]
THEN MRESA_TAC REAL_LT_MUL[`inv (&1- t1):real`;`t3:real`;]
THEN MP_TAC(REAL_ARITH`&0< (inv (&1- t1)) * t3 ==> ~( (inv(&1- t1)) * t3= &0)`)
THEN MP_TAC(REAL_ARITH`&0< (t2') * inv (&1- t1) ==> ~((t2')* inv(&1- t1)= &0)`)
THEN RESA_TAC
THEN ASSUME_TAC(REAL_ARITH`~(&1 = &0)`)
THEN MRESA_TAC COPLANAR_SCALE_ALL [`&1`;`&1`;`(t2')* inv(&1- t1)`;`u - x:real^3`; `v - x:real^3`; `(inv t2' * t2) % (v - x) +(inv t2' * t3) % ((&1 - a) % u + a % w - x):real^3`]
THEN POP_ASSUM (fun th -> ONCE_REWRITE_TAC[SYM(th);])
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%(B+E%(C+D))=A%B+(A*E)%(C+D)`;
VECTOR_ARITH`A%B%C=(A*B)%C`; REAL_ARITH`(t2' * inv (&1 - t1)) * inv t2' * t2= (inv (t2')* t2' ) * inv (&1-t1) * t2`]
THEN REDUCE_ARITH_TAC
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN DISCH_TAC
THEN MRESA_TAC continuous_coplanar_fan[`x:real^3 `;` v:real^3`;` u:real^3`;` w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `a:real`)
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v, (&1 - a) % u + a % w}= {(--x) +x, (--x) +v, (--x) +u, (--x) +(&1 - a) % u + a % w}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, (&1 - a) % u + a % w}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)] )
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN DISCH_TAC
THEN MRESA_TAC continuous_coplanar_fan[`(vec 0):real^3 `;` u-x:real^3`;` v-x:real^3`;` ((&1 - a) % u + a % w)-x:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th`(inv(&1- t1)) * t3:real`)
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`
(&1 - inv (&1 - t1) * t3) % (v - x) +
                        (inv (&1 - t1) * t3) % (((&1 - a) % u + a % w) - x)=
(&1 - inv (&1 - t1) * t3) % (v - x) +
                            (inv (&1 - t1) * t3) % ((&1 - a) % u + a % w - x)`];(*4*)

POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"DICH")
THEN DISCH_TAC
THEN ABBREV_TAC `va=((&1 - a) % u + a % w):real^3`
THEN MP_TAC(REAL_ARITH` ~(t2' = &0) /\ &0 <= t2'==> &0 <t2'`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`  &0 < a==> ~(a = &0) `)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`t2':real`
THEN MRESA_TAC REAL_LT_MUL[`inv (t2'):real`;`t2:real`;]
THEN MRESA_TAC REAL_LT_MUL[`inv (t2'):real`;`t3:real`;]
THEN SUBGOAL_THEN `(v':real^3) IN aff_gt {x} {v,va:real^3}` ASSUME_TAC
THENL(*5*)[

ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC`(&1 - inv t2' * t2 - inv t2' * t3:real)`
THEN EXISTS_TAC`inv t2' * t2:real`
THEN EXISTS_TAC`(inv t2' * t3:real)`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC;(*5*)

REMOVE_THEN "DICH" MP_TAC
THEN REMOVE_THEN "VUT" MP_TAC
THEN DISCH_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_THEN "EM"(fun th-> MRESA1_TAC th `v':real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN MRESA_TAC exists_element_in_half_sapace_fan
[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v':real^3`;`  v:real^3`;`u:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u':real^3)`;
` (v':real^3)`]
THEN MRESA_TAC exists_cut_small_edges_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`;`v':real^3`;`u':real^3`]
THEN DISCH_TAC
THEN REMOVE_ASSUM_TAC
THEN REWRITE_TAC[EMPTY;IN]
THEN POP_ASSUM MP_TAC 
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(REAL_ARITH`&0<t /\ t< &1==> &0< &1-t:real /\ &1-t< &1`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`(&1-t:real)`;`a:real`]
THEN MRESAL_TAC REAL_LT_RMUL[`(&1-t:real)`;`&1`;`a:real`][REAL_ARITH`&1 *a=a`]
THEN REMOVE_THEN "YEU"(fun th-> MRESAL1_TAC th `(&1- t) *a`[VECTOR_ARITH`(&1 - (&1 - t) * a) % u + ((&1 - t) * a) % w=(&1 - t) % ((&1-a)%u+a % w)+t%u`;])
THEN MRESA_TAC aff_gt_subset_aff_ge[`x:real^3`;`v':real^3`;`u':real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`{v',u'} IN (E:(real^3->bool)->bool)`
THEN SET_TAC[]](*5*)](*4*)](*3*);(*2*)



DISJ_CASES_TAC(REAL_ARITH`(t2'= &0) \/ ~(t2':real= &0)`)
THENL(*3*)[

FIND_ASSUM MP_TAC`t1 + t2 + t3 = &1:real`
THEN REWRITE_TAC[REAL_ARITH`t1 + t2 + t3 = &1<=> t1= &1 -t2 -t3`; REAL_ARITH`A+B= &1<=> A= &1-B`]
THEN DISCH_THEN(LABEL_TAC"MAI")
THEN DISCH_THEN (LABEL_TAC"YEU EM")
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(&1 - t2 - t3) % x + t2 % v + t3 % ((&1 - a) % u + a % w) =
 (&1 - &0 - t3') % x + &0% v'+t3' % w'<=> t3' % w'=(t3' - t2 - t3) % x + t2 % v + t3 % ((&1 - a) % u + a % w):real^3 `]
THEN MP_TAC(ISPEC`(t3':real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t3' % w'=(t3' - t2 - t3) % x + t2 % v + t3 % ((&1 - a) % u + a % w):real^3
==> (inv (t3'))%(t3' % w' ) = (inv (t3'))%((t3' - t2 - t3) % x + t2 % v + t3 % ((&1 - a) % u + a % w):real^3)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C+D)=(A%B+A%C+A%D)`;REAL_ARITH`a*(b-c-d)=a*b-a*c-a*d`]
THEN REDUCE_VECTOR_TAC
THEN DISCH_TAC
THEN SUBGOAL_THEN `~coplanar{x:real^3,w':real^3,v:real^3,u:real^3}` ASSUME_TAC
THENL(*4*)[

 POP_ASSUM MP_TAC
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3, w', v, u}= {(--x) +x, (--x) +w', (--x) +v, (--x) +u}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3,w':real^3,v:real^3,u:real^3}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)] THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,C,B}`])
THEN DISCH_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th;VECTOR_ARITH`((&1 - inv t2' * t2 - inv t2' * t3) % x +
                         (inv t2' * t2) % v +
                         (inv t2' * t3) % ((&1 - a) % u + a % w)) -
                        x=(inv t2' * t2) % (v-x) +
                         (inv t2' * t3) % ((&1 - a) % u + a % w-x)`])
THEN MP_TAC(REAL_ARITH`&0< t2 /\ &0< t3 /\ t1+t2+t3= &1==>  &0 < &1- t1 /\ ~(&1- t1 = &0)/\ t2+t3= &1- t1`)
THEN REMOVE_THEN"MAI" MP_TAC
THEN DISCH_TAC
THEN REMOVE_ASSUM_TAC
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ISPEC`&1- (t1:real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`&1- (t1:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2 + t3 = &1 - t1:real
==> (inv ( &1 - t1))*(t2 + t3) = (inv ( &1 - t1))*( &1 - t1:real)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C)= &1<=> A*B = &1 -A*C`]
THEN STRIP_TAC
THEN MP_TAC(REAL_ARITH` ~(t3' = &0) /\ &0 <= t3'==> &0 <t3'`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`  &0 < a==> ~(a = &0) `)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`t3':real`;`inv (&1- t1):real`]
THEN MRESA_TAC REAL_LT_MUL[`inv (&1- t1):real`;`t3:real`;]
THEN MP_TAC(REAL_ARITH`&0< (inv (&1- t1)) * t3 ==> ~( (inv(&1- t1)) * t3= &0)`)
THEN MP_TAC(REAL_ARITH`&0< (t3') * inv (&1- t1) ==> ~((t3')* inv(&1- t1)= &0)`)
THEN RESA_TAC
THEN ASSUME_TAC(REAL_ARITH`~(&1 = &0)`)
THEN MRESA_TAC COPLANAR_SCALE_ALL [`&1`;`&1`;`(t3')* inv(&1- t1)`;`u - x:real^3`; `v - x:real^3`; `(inv t3' * t2) % (v - x) +(inv t3' * t3) % ((&1 - a) % u + a % w - x):real^3`]
THEN POP_ASSUM (fun th -> ONCE_REWRITE_TAC[SYM(th);])
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%(B+E%(C+D))=A%B+(A*E)%(C+D)`;
VECTOR_ARITH`A%B%C=(A*B)%C`; REAL_ARITH`(t2' * inv (&1 - t1)) * inv t2' * t2= (inv (t2')* t2' ) * inv (&1-t1) * t2`]
THEN REDUCE_ARITH_TAC
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN DISCH_TAC
THEN MRESA_TAC continuous_coplanar_fan[`x:real^3 `;` v:real^3`;` u:real^3`;` w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `a:real`)
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v, (&1 - a) % u + a % w}= {(--x) +x, (--x) +v, (--x) +u, (--x) +(&1 - a) % u + a % w}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, (&1 - a) % u + a % w}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)] )
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN DISCH_TAC
THEN MRESA_TAC continuous_coplanar_fan[`(vec 0):real^3 `;` u-x:real^3`;` v-x:real^3`;` ((&1 - a) % u + a % w)-x:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th`(inv(&1- t1)) * t3:real`)
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`
(&1 - inv (&1 - t1) * t3) % (v - x) +
                        (inv (&1 - t1) * t3) % (((&1 - a) % u + a % w) - x)=
(&1 - inv (&1 - t1) * t3) % (v - x) +
                            (inv (&1 - t1) * t3) % ((&1 - a) % u + a % w - x)`];(*4*)


POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"DICH")
THEN DISCH_TAC
THEN ABBREV_TAC `va=((&1 - a) % u + a % w):real^3`
THEN MP_TAC(REAL_ARITH` ~(t3' = &0) /\ &0 <= t3'==> &0 <t3'`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`  &0 < a==> ~(a = &0) `)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`t3':real`
THEN MRESA_TAC REAL_LT_MUL[`inv (t3'):real`;`t2:real`;]
THEN MRESA_TAC REAL_LT_MUL[`inv (t3'):real`;`t3:real`;]
THEN SUBGOAL_THEN `(w':real^3) IN aff_gt {x} {v,va:real^3}` ASSUME_TAC
THENL[(*5*)

ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC`(&1 - inv t3' * t2 - inv t3' * t3:real)`
THEN EXISTS_TAC`inv t3' * t2:real`
THEN EXISTS_TAC`(inv t3' * t3:real)`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC;(*5*)

REMOVE_THEN "DICH" MP_TAC
THEN REMOVE_THEN "VUT" MP_TAC
THEN DISCH_TAC
THEN REMOVE_ASSUM_TAC
THEN FIND_ASSUM MP_TAC`{v',w':real^3} IN E`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN DISCH_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (v':real^3)`;
` (w':real^3)`]  
THEN REMOVE_THEN "EM"(fun th-> MRESA1_TAC th `w':real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN MRESA_TAC exists_element_in_half_sapace_fan
[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`w':real^3`;`  v:real^3`;`u:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u':real^3)`;
` (w':real^3)`]
THEN MRESA_TAC exists_cut_small_edges_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`;`w':real^3`;`u':real^3`]
THEN DISCH_TAC
THEN REMOVE_ASSUM_TAC
THEN REWRITE_TAC[EMPTY;IN]
THEN POP_ASSUM MP_TAC 
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(REAL_ARITH`&0<t /\ t< &1==> &0< &1-t:real /\ &1-t< &1`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`(&1-t:real)`;`a:real`]
THEN MRESAL_TAC REAL_LT_RMUL[`(&1-t:real)`;`&1`;`a:real`][REAL_ARITH`&1 *a=a`]
THEN REMOVE_THEN "YEU"(fun th-> MRESAL1_TAC th `(&1- t) *a`[VECTOR_ARITH`(&1 - (&1 - t) * a) % u + ((&1 - t) * a) % w=(&1 - t) % ((&1-a)%u+a % w)+t%u`;])
THEN MRESA_TAC aff_gt_subset_aff_ge[`x:real^3`;`w':real^3`;`u':real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`{w',u'} IN (E:(real^3->bool)->bool)`
THEN SET_TAC[]](*5*)](*4*);(*3*)


ASM_REWRITE_TAC[]
THEN DISCH_TAC THEN DISCH_THEN(LABEL_TAC"BO")
THEN ABBREV_TAC `va=((&1 - a) % u + a % w):real^3`
THEN MP_TAC(REAL_ARITH` ~(t3' = &0) /\ &0 <= t3'==> &0 <t3'`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH` ~(t2' = &0) /\ &0 <= t2'==> &0 <t2'`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`  &0 < a==> ~(a = &0) `)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`t2':real`
THEN MRESA1_TAC REAL_MUL_LINV`t2':real`
THEN MRESA_TAC REAL_LT_MUL[`inv (t2'):real`;`t2:real`;]
THEN MRESA_TAC REAL_LT_MUL[`inv (t2'):real`;`t3:real`;]
THEN MRESA1_TAC REAL_LT_INV`t3':real`
THEN MRESA1_TAC REAL_MUL_LINV`t3':real`
THEN MRESA_TAC REAL_LT_MUL[`inv (t3'):real`;`t2:real`;]
THEN MRESA_TAC REAL_LT_MUL[`inv (t3'):real`;`t3:real`;]
THEN SUBGOAL_THEN`(x':real^3) IN aff_gt {x} {v',w'}` ASSUME_TAC
THENL(*4*)[

MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v':real^3`;`w':real^3`][IN_ELIM_THM]
THEN ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC`t1':real`
THEN EXISTS_TAC`t2':real`
THEN EXISTS_TAC`t3':real`
THEN ASM_REWRITE_TAC[];(*4*)

SUBGOAL_THEN`(x':real^3) IN aff_ge {x} {v',w'}` ASSUME_TAC
THENL(*5*)[

ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC`t1':real`
THEN EXISTS_TAC`t2':real`
THEN EXISTS_TAC`t3':real`
THEN ASM_REWRITE_TAC[];(*5*)

SUBGOAL_THEN `(x':real^3) IN aff_gt {x} {v,va:real^3}` ASSUME_TAC
THENL(*6*)[

ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3:real`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC;(*6*)

SUBGOAL_THEN `(x':real^3) IN aff_ge {x} {v,va:real^3}` ASSUME_TAC
THENL(*7*)[

MRESA_TAC AFF_GE_1_2[`x:real^3`;`v:real^3`;`va:real^3`]
THEN ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3:real`
THEN MP_TAC(REAL_ARITH`&0< t2==> &0 <= t2`) THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0< t3==> &0 <= t3`) THEN RESA_TAC
THEN ASM_REWRITE_TAC[];(*7*)

FIND_ASSUM MP_TAC`{v',w':real^3} IN E`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN DISCH_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (v':real^3)`;
` (w':real^3)`]  
THEN SUBGOAL_THEN `~collinear {x, x', w':real^3}` ASSUME_TAC
THENL(*8*)[

ASM_REWRITE_TAC[collinear1_fan;]
THEN FIND_ASSUM MP_TAC `~((v':real^3) IN aff{x,w'})`
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff; AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % v' + t3' % w' = u' % x + v'' % w'
<=> t2' % v' = (u'-t1') % x + (v''-t3') % w'`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2' % v' = (u'-t1') % x + (v''-t3') % w':real^3
==> (inv ( t2'))%(t2' % v') = (inv (t2'))%((u'-t1') % x + (v''-t3') % w')
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C= (A*B)%C`;VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC `(inv t2' * (u' - t1':real))`
THEN EXISTS_TAC `(inv t2' * (v'' - t3':real))`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv t2' * (u' - t1') + inv t2' * (v'' - t3')=inv t2'* (t2'+ (u'+v'') - (t1'+t2'+t3'))`;REAL_ARITH`t2'+ &1 - &1=t2'`];(*8*)




SUBGOAL_THEN `~collinear {x, x', v':real^3}` ASSUME_TAC
THENL(*9*)[
ASM_REWRITE_TAC[collinear1_fan;]
THEN FIND_ASSUM MP_TAC `~((w':real^3) IN aff{x,v'})`
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff; AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % v' + t3' % w' = u' % x + v'' % v'
<=> t3' % w' = (u'-t1') % x + (v''-t2') % v'`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t3' % w' = (u'-t1') % x + (v''-t2') % v':real^3
==> (inv ( t3'))%(t3' % w') = (inv (t3'))%((u'-t1') % x + (v''-t2') % v')
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C= (A*B)%C`;VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC `(inv t3' * (u' - t1':real))`
THEN EXISTS_TAC `(inv t3' * (v'' - t2':real))`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv t3' * (u' - t1') + inv t3' * (v'' - t2')=inv t3'* (t3'+ (u'+v'') - (t1'+t2'+t3'))`;REAL_ARITH`t3'+ &1 - &1=t3'`];(*9*)




SUBGOAL_THEN`~collinear {x, x', va:real^3}` ASSUME_TAC
THENL(*10*)[
REMOVE_THEN "BO" MP_TAC
THEN DISCH_TAC THEN REMOVE_ASSUM_TAC
THEN ASM_REWRITE_TAC[collinear1_fan;]
THEN FIND_ASSUM MP_TAC`~(va IN aff {x, v:real^3})`
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1 % x + t2 % v + t3 % (va) =
      u' % x + v' % (va)<=>
(&1-(t1+t2+t3)) % x  + (v'-t3) % (va-x) =
      (&1-(u'+v') ) % x + t2 % (v-x)`;REAL_ARITH`a-a= &0`]
THEN REDUCE_VECTOR_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(v'' - t3 = &0) \/ ~(v'' - (t3:real)= &0)`)
THENL(*11*)[

ASM_REWRITE_TAC[VECTOR_ARITH`&0 %A=B<=> B= vec 0`;VECTOR_MUL_EQ_0]
THEN MP_TAC(REAL_ARITH`&0< t2==> ~(t2 = &0)`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A-B= vec 0<=> A=B`];(*11*)

MRESA1_TAC REAL_MUL_LINV `(v''-t3:real)`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(v'' - t3) % (va- x) = t2 % (v - x:real^3) ==> (inv (v'' -t3)) % ((v'' - t3) % (va - x)) = (inv (v'' -t3)) % ( t2 % (v - x))`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th]
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(A%B%C=(A*B)%C:real^3)`])
THEN REWRITE_TAC[VECTOR_ARITH`&1 % (va - x) = (inv (v' - t3) * t2) % (v - x)<=>
va  = (&1-(inv (v' - t3) * t2))%x +(inv (v' - t3) * t2) % v:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`&1-(inv (v'' - t3) * (t2:real))`
THEN EXISTS_TAC`(inv (v'' - t3) * (t2:real))`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC](*11*);(*10*)


SUBGOAL_THEN `~collinear {x, x', v:real^3}` ASSUME_TAC
THENL(*11*)[
REMOVE_THEN "BO" MP_TAC
THEN DISCH_TAC
THEN REMOVE_ASSUM_TAC
THEN ASM_REWRITE_TAC[collinear1_fan]
THEN FIND_ASSUM MP_TAC `~(va IN aff {x, v:real^3})`
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff; AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % v' + t3' % w' = u' % x + v'' % v'
<=> t3' % w' = (u'-t1') % x + (v''-t2') % v'`]
THEN MP_TAC(REAL_ARITH`&0< t3 ==> ~(t3= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`t3:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t3 % va = (u'-t1) % x + (v''-t2) % v:real^3
==> (inv ( t3))%(t3 % va) = (inv (t3))%((u'-t1) % x + (v''-t2) % v)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C= (A*B)%C`;VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC `(inv t3 * (u' - t1:real))`
THEN EXISTS_TAC `(inv t3 * (v'' - t2:real))`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv t3' * (u' - t1') + inv t3' * (v'' - t2')=inv t3'* (t3'+ (u'+v'') - (t1'+t2'+t3'))`;REAL_ARITH`t3'+ &1 - &1=t3'`];(*11*)

ABBREV_TAC`an=t1' % x + t2' % v' + t3' % w':real^3`
THEN REMOVE_THEN "VUT1" MP_TAC
THEN REMOVE_THEN "VUT1" MP_TAC
THEN REMOVE_THEN "BO" MP_TAC
THEN REMOVE_THEN "VUT" MP_TAC
THEN DISCH_TAC THEN DISCH_TAC THEN DISCH_TAC THEN DISCH_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN DISJ_CASES_TAC(SET_RULE`(&0 < azim x (x':real^3) v w' /\
      azim x (x') v w' < pi)\/ ~(&0 < azim x (x') v w' /\
      azim x (x') v w' < pi)`)

THENL(*12*)[

MRESA_TAC exists_cut_small_edges_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`;`x':real^3`;`w':real^3`]
THEN REWRITE_TAC[EMPTY;IN]
THEN MP_TAC(REAL_ARITH`&0<t /\ t< &1==> &0< &1-t:real /\ &1-t< &1`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`(&1-t:real)`;`a:real`]
THEN MRESAL_TAC REAL_LT_RMUL[`(&1-t:real)`;`&1`;`a:real`][REAL_ARITH`&1 *a=a`]
THEN REMOVE_THEN "YEU"(fun th-> MRESAL1_TAC th `(&1- t) *a`[VECTOR_ARITH`(&1 - (&1 - t) * a) % u + ((&1 - t) * a) % w=(&1 - t) % ((&1-a)%u+a % w)+t%u`;])
THEN MRESA_TAC aff_gt1_subset_aff_ge[`x:real^3`;`v':real^3`;`w':real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`{v',w'} IN (E:(real^3->bool)->bool)`
THEN SET_TAC[];(*12*)

ASSUME_TAC(ISPECL[`x:real^3`;`v':real^3`;`w':real^3`;`x':real^3`]azim)
THEN ASSUME_TAC(PI_WORKS)
THEN MP_TAC(REAL_ARITH`~(&0 < azim x x' v w' /\ azim x x' v w' < pi)/\
&0 <= azim x x' v w' /\ &0< pi==>   pi< azim x x' v w' \/(azim x x' v w'= &0 \/  azim x x' v w'=pi) `)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*13*)(*3SUBGOAL*)[

POP_ASSUM MP_TAC
THEN MRESA_TAC aff_gt2_subset_aff_ge[`x:real^3`;`v':real^3`;`w':real^3 `;`x':real^3`]
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN DISCH_TAC
THEN MP_TAC(REAL_ARITH`azim x x' v' w' < azim x x' v w'==>azim x x' v' w' <= azim x x' v w'`)
THEN RESA_TAC
THEN MRESA_TAC sum5_azim_fan[`x:real^3`;`x':real^3`;`v:real^3`;`v':real^3`;`w':real^3`]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC aff_gt2_subset_aff_ge[`x:real^3`;`v':real^3`;`w':real^3 `;`x':real^3`]
THEN DISCH_TAC THEN DISCH_TAC
THEN MP_TAC(REAL_ARITH`pi < azim x x' v w' /\ azim x x' v w' = azim x x' v v' + pi
/\  azim x x' v w' < &2 * pi==> &0 <azim x x' v v' /\ azim x x' v v' < pi`)
THEN RESA_TAC
THEN MRESA_TAC exists_cut_small_edges_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`;`x':real^3`;`v':real^3`]
THEN REWRITE_TAC[EMPTY;IN]
THEN MP_TAC(REAL_ARITH`&0<t /\ t< &1==> &0< &1-t:real /\ &1-t< &1`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`(&1-t:real)`;`a:real`]
THEN MRESAL_TAC REAL_LT_RMUL[`(&1-t:real)`;`&1`;`a:real`][REAL_ARITH`&1 *a=a`]
THEN REMOVE_THEN "YEU"(fun th-> MRESAL1_TAC th `(&1- t) *a`[VECTOR_ARITH`(&1 - (&1 - t) * a) % u + ((&1 - t) * a) % w=(&1 - t) % ((&1-a)%u+a % w)+t%u`;])
THEN SUBGOAL_THEN ` aff_ge {x} {w', v':real^3}=aff_ge {x} {v', w'}` ASSUME_TAC
THENL(*14*)[
REWRITE_TAC[SET_RULE`{A,B}={B,A}`];(*14*)

MRESA_TAC aff_gt1_subset_aff_ge[`x:real^3`;`w':real^3`;`v':real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC 
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`{v',w'} IN (E:(real^3->bool)->bool)`
THEN SET_TAC[]](*14*);(*13*)

MRESA_TAC AZIM_EQ_0[`x:real^3`;`x':real^3`;`v:real^3`;`w':real^3`]
THEN MRESA_TAC AZIM_EQ_0_ALT[`x:real^3`;`x':real^3`;`v:real^3`;`w':real^3`]
THEN MRESA_TAC AZIM_EQ_0[`x:real^3`;`x':real^3`;`w':real^3`;`v:real^3`]
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`x':real^3`;`w':real^3`;`v:real^3`]
THEN DISJ_CASES_TAC(SET_RULE`w' IN aff_gt {x} {v,va:real^3}\/ ~(w' IN aff_gt {x} {v,va:real^3})`)
THENL(*14*)[
MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`(&1-a)%u +a%w:real^3`][IN_ELIM_THM]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"DICH CHUYEN")
THEN DISCH_TAC
THEN REMOVE_THEN "DICH CHUYEN"(fun th->MP_TAC(th) THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN REMOVE_ASSUM_TAC
 THEN ASSUME_TAC(th))
THEN STRIP_TAC 
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`w' = t1'' % x + t2'' % v + t3'' % va<=>
 ((t1''+t2''+t3'')- &1) % x + t2'' % (v-x) + t3'' % (va-x)= w' -x:real^3`;REAL_ARITH`&1- &1 = &0`]
THEN REDUCE_VECTOR_TAC
THEN MP_TAC(REAL_ARITH`&0< t2'' /\ &0< t3'' /\ t1''+t2''+t3''= &1==>  &0 < &1- t1'' /\ ~(&1- t1'' = &0)/\ t2''+t3''= &1- t1''`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ISPEC`&1- (t1'':real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`&1- (t1'':real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2'' + t3'' = &1 - t1'':real
==> (inv ( &1 - t1''))*(t2'' + t3'') = (inv ( &1 - t1''))*( &1 - t1'':real)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C)= &1<=> &1- A*C = A*B`]
THEN STRIP_TAC
THEN MRESA_TAC REAL_LT_MUL[`inv(&1- t1'':real)`;`t3'':real`]
THEN MP_TAC(REAL_ARITH`&0<inv(&1- t1'':real)*(t3'':real)==> ~(inv(&1- t1'':real)*(t3'':real)= &0)`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0<inv(&1- t1'':real)==> ~(inv(&1- t1'':real)= &0)`)
THEN RESA_TAC
THEN ASSUME_TAC(REAL_ARITH`~(&1= &0)`)
THEN STRIP_TAC
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC continuous_coplanar_fan[`x:real^3 `;` v:real^3`;` u:real^3`;` w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `a:real`)
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v, (&1 - a) % u + a % w}= {(--x) +x, (--x) +v, (--x) +u, (--x) +(&1 - a) % u + a % w}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, (&1 - a) % u + a % w}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)] )
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN DISCH_TAC
THEN MRESA_TAC continuous_coplanar_fan[`(vec 0):real^3 `;` u-x:real^3`;` v-x:real^3`;` ((&1 - a) % u + a % w)-x:real^3`]
THEN POP_ASSUM (fun th-> MRESAL1_TAC th`(inv(&1- t1'')) * t3'':real`[VECTOR_ARITH`(A*B)%X+(A*C)%Y=A%(B%X+C%Y)`])
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC COPLANAR_SCALE_ALL [`&1`;`&1`;`inv(&1- t1'')`;`u - x:real^3`; `v - x:real^3`; `w'-x:real^3`]
[VECTOR_ARITH`&1 %A=A`]
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v,w'}= {(--x) +x, (--x) +u, (--x) +v, (--x) + w'}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, w'}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,C,B}`]
THEN DISCH_TAC
THEN REMOVE_THEN "EM"(fun th-> MRESA1_TAC th `w':real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN MRESA_TAC exists_element_in_half_sapace_fan
[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`w':real^3`;`  v:real^3`;`u:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u':real^3)`;
` (w':real^3)`]
THEN MRESA_TAC exists_cut_small_edges_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`;`w':real^3`;`u':real^3`]
THEN REWRITE_TAC[EMPTY;IN]
THEN MP_TAC(REAL_ARITH`&0<t /\ t< &1==> &0< &1-t:real /\ &1-t< &1`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`(&1-t:real)`;`a:real`]
THEN MRESAL_TAC REAL_LT_RMUL[`(&1-t:real)`;`&1`;`a:real`][REAL_ARITH`&1 *a=a`]
THEN REMOVE_THEN "YEU"(fun th-> MRESAL1_TAC th `(&1- t) *a`[VECTOR_ARITH`(&1 - (&1 - t) * a) % u + ((&1 - t) * a) % w=(&1 - t) % ((&1-a)%u+a % w)+t%u`;])
THEN MRESA_TAC aff_gt_subset_aff_ge[`x:real^3`;`w':real^3`;`u':real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`{w',u'} IN (E:(real^3->bool)->bool)`
THEN SET_TAC[];(*14*)

DISJ_CASES_TAC(SET_RULE`v' IN aff_gt {x} {v,va:real^3}\/ ~(v' IN aff_gt {x} {v,va:real^3})`)
THENL(*15*)[

MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`(&1-a)%u +a%w:real^3`][IN_ELIM_THM]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"DICH CHUYEN")
THEN DISCH_TAC
THEN REMOVE_THEN "DICH CHUYEN"(fun th->MP_TAC(th) THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN REMOVE_ASSUM_TAC
 THEN ASSUME_TAC(th))
THEN STRIP_TAC 
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`v' = t1'' % x + t2'' % v + t3'' % va<=>
 ((t1''+t2''+t3'')- &1) % x + t2'' % (v-x) + t3'' % (va-x)= v' -x:real^3`;REAL_ARITH`&1- &1 = &0`]
THEN REDUCE_VECTOR_TAC
THEN MP_TAC(REAL_ARITH`&0< t2'' /\ &0< t3'' /\ t1''+t2''+t3''= &1==>  &0 < &1- t1'' /\ ~(&1- t1'' = &0)/\ t2''+t3''= &1- t1''`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ISPEC`&1- (t1'':real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`&1- (t1'':real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2'' + t3'' = &1 - t1'':real
==> (inv ( &1 - t1''))*(t2'' + t3'') = (inv ( &1 - t1''))*( &1 - t1'':real)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C)= &1<=> &1- A*C = A*B`]
THEN STRIP_TAC
THEN MRESA_TAC REAL_LT_MUL[`inv(&1- t1'':real)`;`t3'':real`]
THEN MP_TAC(REAL_ARITH`&0<inv(&1- t1'':real)*(t3'':real)==> ~(inv(&1- t1'':real)*(t3'':real)= &0)`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0<inv(&1- t1'':real)==> ~(inv(&1- t1'':real)= &0)`)
THEN RESA_TAC
THEN ASSUME_TAC(REAL_ARITH`~(&1= &0)`)
THEN STRIP_TAC
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC continuous_coplanar_fan[`x:real^3 `;` v:real^3`;` u:real^3`;` w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `a:real`)
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v, (&1 - a) % u + a % w}= {(--x) +x, (--x) +v, (--x) +u, (--x) +(&1 - a) % u + a % w}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, (&1 - a) % u + a % w}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)] )
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN DISCH_TAC
THEN MRESA_TAC continuous_coplanar_fan[`(vec 0):real^3 `;` u-x:real^3`;` v-x:real^3`;` ((&1 - a) % u + a % w)-x:real^3`]
THEN POP_ASSUM (fun th-> MRESAL1_TAC th`(inv(&1- t1'')) * t3'':real`[VECTOR_ARITH`(A*B)%X+(A*C)%Y=A%(B%X+C%Y)`])
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC COPLANAR_SCALE_ALL [`&1`;`&1`;`inv(&1- t1'')`;`u - x:real^3`; `v - x:real^3`; `v'-x:real^3`]
[VECTOR_ARITH`&1 %A=A`]
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v,v'}= {(--x) +x, (--x) +u, (--x) +v, (--x) + v'}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, v'}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,C,B}`]
THEN DISCH_TAC
THEN REMOVE_THEN "EM"(fun th-> MRESA1_TAC th `v':real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN MRESA_TAC exists_element_in_half_sapace_fan
[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v':real^3`;`  v:real^3`;`u:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u':real^3)`;
` (v':real^3)`]
THEN MRESA_TAC exists_cut_small_edges_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`;`v':real^3`;`u':real^3`]
THEN REWRITE_TAC[EMPTY;IN]
THEN MP_TAC(REAL_ARITH`&0<t /\ t< &1==> &0< &1-t:real /\ &1-t< &1`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`(&1-t:real)`;`a:real`]
THEN MRESAL_TAC REAL_LT_RMUL[`(&1-t:real)`;`&1`;`a:real`][REAL_ARITH`&1 *a=a`]
THEN REMOVE_THEN "YEU"(fun th-> MRESAL1_TAC th `(&1- t) *a`[VECTOR_ARITH`(&1 - (&1 - t) * a) % u + ((&1 - t) * a) % w=(&1 - t) % ((&1-a)%u+a % w)+t%u`;])
THEN MRESA_TAC aff_gt_subset_aff_ge[`x:real^3`;`v':real^3`;`u':real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`{v',u'} IN (E:(real^3->bool)->bool)`
THEN SET_TAC[];(*15*)

SUBGOAL_THEN`~(w' IN aff_gt {x} { x',v:real^3})` ASSUME_TAC
THENL(*16*)[
REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN MRESA_TAC aff_gt3_subset_aff_gt[`x:real^3`;`v:real^3`;`va:real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
THEN SET_TAC[];(*16*)

MP_TAC(ISPECL[`x:real^3`;`x':real^3`;`v:real^3`;`w':real^3`] decomposition_planar_by_angle_fan)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*17*)[
MRESA_TAC aff_gt1_subset_aff_ge[`x:real^3`;`v':real^3`;`w':real^3`;`x':real^3`]
THEN MP_TAC(SET_RULE`v IN aff_gt {x} {x', w':real^3} /\ aff_gt {x} {x', w'} SUBSET aff_ge {x} {v', w'}
==>v IN aff_ge {x} {v', w':real^3}`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`; `v:real^3`; `u:real^3`; `v':real^3`;`w':real^3`]properties_of_fan7)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*18*)[
MRESA_TAC aff_gt2_subset_aff_ge[`x:real^3`;`v':real^3`; `w':real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th-> ASM_REWRITE_TAC[SYM(th)])
THEN MP_TAC(PI_WORKS)
THEN REAL_ARITH_TAC;(*18*)

MRESA_TAC aff_gt2_subset_aff_ge[`x:real^3`;`v':real^3`; `w':real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th-> ASM_REWRITE_TAC[SYM(th)])
THEN MRESA_TAC aff_gt2_subset_aff_ge[`x:real^3`;`v:real^3`; `va:real^3`;`x':real^3`]
THEN DISCH_TAC
THEN MRESA_TAC AZIM_EQ_PI_SYM [`x:real^3`;`x':real^3`; `v':real^3`;`v:real^3`]
THEN ASSUME_TAC(REAL_ARITH`pi <= pi`)
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`x':real^3`;`v:real^3`;`va:real^3`;`v':real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`A=A+B <=> B= &0`]
THEN DISCH_TAC
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`x':real^3`;`va:real^3`;`v':real^3`]
THEN MP_TAC(ISPECL[`x:real^3`;`x':real^3`;`v':real^3`;`va:real^3`] decomposition_planar_by_angle_fan)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*19*)[
MRESA_TAC aff_gt1_subset_aff_gt[`x:real^3`;`v:real^3`; `va:real^3`;`x':real^3`]
THEN ASM_TAC THEN SET_TAC[];(*19*)

MRESA_TAC aff_ge1_subset_aff_ge[`x:real^3`;`w':real^3`; `v':real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN STRIP_TAC THEN STRIP_TAC
THEN MP_TAC(SET_RULE`va IN aff_ge {x} {v', x':real^3} /\ aff_ge {x} {v', x'} SUBSET aff_ge {x} {v', w'}
==>va IN aff_ge {x} {v', w':real^3}`)
THEN RESA_TAC
THEN SUBGOAL_THEN `va IN aff_ge {x} {u,w:real^3}` ASSUME_TAC
THENL(*20*)[

MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`u:real^3`]
THEN MRESAL_TAC AFF_GE_1_2[`x:real^3`;`u:real^3`;`w:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1 -a:real`
THEN EXISTS_TAC`a:real`
THEN MP_TAC(REAL_ARITH`&0< a /\ a  < &1 ==> &0 <= &1 - a /\ &0 <= a`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC;(*20*)


MP_TAC(SET_RULE`va IN aff_ge {x} {v', w':real^3} /\ va IN aff_ge {x} {u, w} 
==>va IN  aff_ge {x} {u, w} INTER aff_ge {x} {v', w':real^3} `)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC `FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN;fan7]
THEN STRIP_TAC
THEN POP_ASSUM  (fun th-> MP_TAC(ISPECL[`{(u:real^3),(w:real^3)}`;`{(v':real^3),(w':real^3)}`]th)) 
  THEN ASM_REWRITE_TAC[UNION;IN_ELIM_THM;]
THEN DISCH_TAC THEN ASM_REWRITE_TAC[]
THEN DISJ_CASES_TAC(SET_RULE`{u, w} INTER {v', w'}={u,w:real^3}\/ ~({u, w} INTER {v', w'}={u,w:real^3})`)
THENL(*21*)[

DISCH_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`u:real^3`]
THEN REMOVE_ASSUM_TAC
THEN MP_TAC(SET_RULE`{u, w} INTER {v', w'}={u,w:real^3}/\ ~(w=u) /\ ~(v'=w') ==> {v', w'}={u,w} `)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC `v IN aff_ge {x} {v', w':real^3}`
THEN POP_ASSUM(fun th->REWRITE_TAC[th])
THEN MRESAL_TAC aff_ge_inter_aff_ge[`x:real^3`;`u:real^3`;`w:real^3`][INTER;IN_ELIM_THM]
THEN STRIP_TAC
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`u:real^3`;`v:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[]
THEN DISCH_TAC
THEN MRESA_TAC AZIM_EQ_0_PI_IMP_COPLANAR[`x:real^3`;`u:real^3`;`v:real^3`;`w:real^3`]
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];(*21*)


MP_TAC(SET_RULE`~({u, w} INTER {v', w'} = {u, w:real^3}) ==> {u, w} INTER {v', w'} PSUBSET {u, w} `)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`{u, w} INTER {v', w'} PSUBSET {u, w:real^3} ==> {u, w} INTER {v', w'} SUBSET {u}\/  {u, w} INTER {v', w'} SUBSET {w} `)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*22*)[

MRESAL_TAC AFF_GE_MONO_RIGHT [`{x:real^3}`;`{u, w} INTER {v', w':real^3}`; `{u:real^3}`][DISJOINT;SET_RULE`{x} INTER {u} = {}<=> ~(x=u:real^3)`]
THEN STRIP_TAC
THEN MRESAL_TAC AFF_GE_SUBSET_AFFINE_HULL[`{x:real^3}`;`{u:real^3}`][SET_RULE`{x} UNION {u}={x,u}`;GSYM aff]
THEN MP_TAC(SET_RULE`aff_ge {x:real^3} ({u, w} INTER {v', w'}) SUBSET aff_ge {x} {u}/\ aff_ge {x} {u} SUBSET aff {x, u}/\ va IN aff_ge {x} ({u, w} INTER {v', w'})==> va IN aff {x,u}`)
THEN RESA_TAC
THEN REWRITE_TAC[IN;EMPTY]
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`u:real^3`]
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC"va"
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - a) % u + a % w = u' % x + v'' % u
<=> a % w = u' % x + (v''+a- &1) % u`]
THEN MRESA1_TAC REAL_MUL_LINV`a:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`a % w = u' % x + (v'' + a - &1) % u:real^3 ==> (inv (a))%(a % w ) = (inv (a))%( u' % x + (v'' + a - &1) % u)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`(inv a * u'):real`
THEN EXISTS_TAC`(inv a * (v'' + a - &1)):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv a * u' + inv a * (v'' + a - &1) =inv a * ( a+ (u' + v'') - &1)`;REAL_ARITH`A+ &1- &1=A`];(*22*)

STRIP_TAC
THEN FIND_ASSUM MP_TAC`{u,w:real^3} IN E`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN STRIP_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` u:real^3`;`w:real^3`]
THEN MRESAL_TAC AFF_GE_MONO_RIGHT [`{x:real^3}`;`{u, w} INTER {v', w':real^3}`; `{w:real^3}`][DISJOINT;SET_RULE`{x} INTER {w} = {}<=> ~(x=w:real^3)`]
THEN MRESAL_TAC AFF_GE_SUBSET_AFFINE_HULL[`{x:real^3}`;`{w:real^3}`][SET_RULE`{x} UNION {u}={x,u}`;GSYM aff]
THEN MP_TAC(SET_RULE`aff_ge {x:real^3} ({u, w} INTER {v', w'}) SUBSET aff_ge {x} {w}/\ aff_ge {x} {w} SUBSET aff {x, w}/\ va IN aff_ge {x} ({u, w} INTER {v', w'})==> va IN aff {x,w}`)
THEN RESA_TAC
THEN REWRITE_TAC[IN;EMPTY]
THEN POP_ASSUM MP_TAC 
THEN FIND_ASSUM MP_TAC `~(u IN aff {x, w:real^3})`
THEN REWRITE_TAC[]
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC"va"
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - a) % u + a % w = u' % x + v'' % w
<=> (&1-a )% u = u' % x + (v''-a) % w`]
THEN MP_TAC(REAL_ARITH`a< &1==> ~(&1-a= &0)` )THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`&1-a:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(&1-a )% u = u' % x + (v''-a) % w:real^3 ==> (inv (&1-a))%((&1-a )% u ) = (inv (&1-a))%( u' % x + (v''-a) % w)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`(inv (&1-a) * u'):real`
THEN EXISTS_TAC`(inv (&1-a) * (v'' - a)):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (&1 - a) * u' + inv (&1 - a) * (v'' - a)=inv (&1 - a) * ((u' + v'') - a)`]](*22*)](*21*)](*20*)](*19*)](*18*);(*17*)




POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN DISCH_TAC
THEN MRESA_TAC aff_ge_subset_aff_gt_union_aff_ge[`(x:real^3)`;`(v:real^3)`;`(x':real^3)`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[]
THEN DISCH_TAC
THEN MP_TAC(SET_RULE`aff_ge {x} {v, x':real^3} SUBSET aff_gt {x, v} {x'} UNION aff_ge {x} {v} /\ w' IN aff_ge {x} {v, x'} ==> w' IN aff_gt {x, v} {x'} UNION aff_ge {x} {v}`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC aff_gt_inter_aff_gt[`(x:real^3)`;`(x':real^3)`;`(v:real^3)`]
THEN ASM_REWRITE_TAC[INTER;IN_ELIM_THM;UNION]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MP_TAC(ISPECL[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`; `w':real^3`; `v':real^3`; `v:real^3`;`u:real^3`]properties1_of_fan7)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> MP_TAC(SYM(th)))
THEN DISCH_TAC
THEN MRESA_TAC point_in_aff_ge[`(x:real^3)`;`(v':real^3)`;`(w':real^3)`]
THEN SUBGOAL_THEN`v IN aff_ge {x} {v', w':real^3}`ASSUME_TAC
THENL(*18*)[
POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[];(*18*)
POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"NHAT")
THEN DISCH_TAC
THEN REMOVE_THEN "NHAT" MP_TAC
THEN DISCH_TAC
THEN MRESA_TAC aff_gt2_subset_aff_ge[`x:real^3`;`v':real^3`; `w':real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th-> ASM_REWRITE_TAC[SYM(th)])
THEN MRESA_TAC aff_gt2_subset_aff_ge[`x:real^3`;`v:real^3`; `va:real^3`;`x':real^3`]
THEN DISCH_TAC
THEN MRESA_TAC AZIM_EQ_PI_SYM [`x:real^3`;`x':real^3`; `v':real^3`;`v:real^3`]
THEN ASSUME_TAC(REAL_ARITH`pi <= pi`)
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`x':real^3`;`v:real^3`;`va:real^3`;`v':real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`A=A+B <=> B= &0`]
THEN DISCH_TAC
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`x':real^3`;`va:real^3`;`v':real^3`]
THEN MP_TAC(ISPECL[`x:real^3`;`x':real^3`;`v':real^3`;`va:real^3`] decomposition_planar_by_angle_fan)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*19*)[

MRESA_TAC aff_gt1_subset_aff_gt[`x:real^3`;`v:real^3`; `va:real^3`;`x':real^3`]
THEN ASM_TAC THEN SET_TAC[];(*19*)

MRESA_TAC aff_ge1_subset_aff_ge[`x:real^3`;`w':real^3`; `v':real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN STRIP_TAC THEN STRIP_TAC
THEN MP_TAC(SET_RULE`va IN aff_ge {x} {v', x':real^3} /\ aff_ge {x} {v', x'} SUBSET aff_ge {x} {v', w'}
==>va IN aff_ge {x} {v', w':real^3}`)
THEN RESA_TAC
THEN SUBGOAL_THEN `va IN aff_ge {x} {u,w:real^3}` ASSUME_TAC
THENL(*20*)[

MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`u:real^3`]
THEN MRESAL_TAC AFF_GE_1_2[`x:real^3`;`u:real^3`;`w:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1 -a:real`
THEN EXISTS_TAC`a:real`
THEN MP_TAC(REAL_ARITH`&0< a /\ a  < &1 ==> &0 <= &1 - a /\ &0 <= a`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC;(*20*)

MP_TAC(SET_RULE`va IN aff_ge {x} {v', w':real^3} /\ va IN aff_ge {x} {u, w} 
==>va IN  aff_ge {x} {u, w} INTER aff_ge {x} {v', w':real^3} `)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC `FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN;fan7]
THEN STRIP_TAC
THEN POP_ASSUM  (fun th-> MP_TAC(ISPECL[`{(u:real^3),(w:real^3)}`;`{(v':real^3),(w':real^3)}`]th)) 
  THEN ASM_REWRITE_TAC[UNION;IN_ELIM_THM;]
THEN DISCH_TAC THEN ASM_REWRITE_TAC[]
THEN DISJ_CASES_TAC(SET_RULE`{u, w} INTER {v', w'}={u,w:real^3}\/ ~({u, w} INTER {v', w'}={u,w:real^3})`)
THENL(*21*)[

DISCH_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`u:real^3`]
THEN REMOVE_ASSUM_TAC
THEN MP_TAC(SET_RULE`{u, w} INTER {v', w'}={u,w:real^3}/\ ~(w=u) /\ ~(v'=w') ==> {v', w'}={u,w} `)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC `v IN aff_ge {x} {v', w':real^3}`
THEN POP_ASSUM(fun th->REWRITE_TAC[th])
THEN MRESAL_TAC aff_ge_inter_aff_ge[`x:real^3`;`u:real^3`;`w:real^3`][INTER;IN_ELIM_THM]
THEN STRIP_TAC
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`u:real^3`;`v:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[]
THEN DISCH_TAC
THEN MRESA_TAC AZIM_EQ_0_PI_IMP_COPLANAR[`x:real^3`;`u:real^3`;`v:real^3`;`w:real^3`]
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];(*21*)

MP_TAC(SET_RULE`~({u, w} INTER {v', w'} = {u, w:real^3}) ==> {u, w} INTER {v', w'} PSUBSET {u, w} `)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`{u, w} INTER {v', w'} PSUBSET {u, w:real^3} ==> {u, w} INTER {v', w'} SUBSET {u}\/  {u, w} INTER {v', w'} SUBSET {w} `)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*22*)[

MRESAL_TAC AFF_GE_MONO_RIGHT [`{x:real^3}`;`{u, w} INTER {v', w':real^3}`; `{u:real^3}`][DISJOINT;SET_RULE`{x} INTER {u} = {}<=> ~(x=u:real^3)`]
THEN STRIP_TAC
THEN MRESAL_TAC AFF_GE_SUBSET_AFFINE_HULL[`{x:real^3}`;`{u:real^3}`][SET_RULE`{x} UNION {u}={x,u}`;GSYM aff]
THEN MP_TAC(SET_RULE`aff_ge {x:real^3} ({u, w} INTER {v', w'}) SUBSET aff_ge {x} {u}/\ aff_ge {x} {u} SUBSET aff {x, u}/\ va IN aff_ge {x} ({u, w} INTER {v', w'})==> va IN aff {x,u}`)
THEN RESA_TAC
THEN REWRITE_TAC[IN;EMPTY]
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`u:real^3`]
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC"va"
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - a) % u + a % w = u' % x + v'' % u
<=> a % w = u' % x + (v''+a- &1) % u`]
THEN MRESA1_TAC REAL_MUL_LINV`a:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`a % w = u' % x + (v'' + a - &1) % u:real^3 ==> (inv (a))%(a % w ) = (inv (a))%( u' % x + (v'' + a - &1) % u)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`(inv a * u'):real`
THEN EXISTS_TAC`(inv a * (v'' + a - &1)):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv a * u' + inv a * (v'' + a - &1) =inv a * ( a+ (u' + v'') - &1)`;REAL_ARITH`A+ &1- &1=A`];(*22*)


STRIP_TAC
THEN FIND_ASSUM MP_TAC`{u,w:real^3} IN E`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN STRIP_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` u:real^3`;`w:real^3`]
THEN MRESAL_TAC AFF_GE_MONO_RIGHT [`{x:real^3}`;`{u, w} INTER {v', w':real^3}`; `{w:real^3}`][DISJOINT;SET_RULE`{x} INTER {w} = {}<=> ~(x=w:real^3)`]
THEN MRESAL_TAC AFF_GE_SUBSET_AFFINE_HULL[`{x:real^3}`;`{w:real^3}`][SET_RULE`{x} UNION {u}={x,u}`;GSYM aff]
THEN MP_TAC(SET_RULE`aff_ge {x:real^3} ({u, w} INTER {v', w'}) SUBSET aff_ge {x} {w}/\ aff_ge {x} {w} SUBSET aff {x, w}/\ va IN aff_ge {x} ({u, w} INTER {v', w'})==> va IN aff {x,w}`)
THEN RESA_TAC
THEN REWRITE_TAC[IN;EMPTY]
THEN POP_ASSUM MP_TAC 
THEN FIND_ASSUM MP_TAC `~(u IN aff {x, w:real^3})`
THEN REWRITE_TAC[]
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC"va"
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - a) % u + a % w = u' % x + v'' % w
<=> (&1-a )% u = u' % x + (v''-a) % w`]
THEN MP_TAC(REAL_ARITH`a< &1==> ~(&1-a= &0)` )THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`&1-a:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(&1-a )% u = u' % x + (v''-a) % w:real^3 ==> (inv (&1-a))%((&1-a )% u ) = (inv (&1-a))%( u' % x + (v''-a) % w)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`(inv (&1-a) * u'):real`
THEN EXISTS_TAC`(inv (&1-a) * (v'' - a)):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (&1 - a) * u' + inv (&1 - a) * (v'' - a)=inv (&1 - a) * ((u' + v'') - a)`]](*22*)](*21*)](*20*)](*19*)](*18*)](*17*)](*16*)](*15*)](*14*);(*13*)



POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"CUOI")
THEN MRESA_TAC aff_gt2_subset_aff_ge[`x:real^3`;`v':real^3`;`w':real^3 `;`x':real^3`]
THEN ASSUME_TAC(REAL_ARITH`pi<= pi`)
THEN MRESA_TAC sum5_azim_fan[`x:real^3`;`x':real^3`;`v:real^3`;`v':real^3`;`w':real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`A=B+A<=>B= &0`]
THEN DISCH_TAC
THEN MRESA_TAC AZIM_EQ_0[`x:real^3`;`x':real^3`;`v:real^3`;`v':real^3`]
THEN MRESA_TAC AZIM_EQ_0_ALT[`x:real^3`;`x':real^3`;`v:real^3`;`v':real^3`]
THEN MRESA_TAC AZIM_EQ_0[`x:real^3`;`x':real^3`;`v':real^3`;`v:real^3`]
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`x':real^3`;`v':real^3`;`v:real^3`]
THEN DISJ_CASES_TAC(SET_RULE`w' IN aff_gt {x} {v,va:real^3}\/ ~(w' IN aff_gt {x} {v,va:real^3})`)
THENL(*14*)[

MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`(&1-a)%u +a%w:real^3`][IN_ELIM_THM]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"DICH CHUYEN")
THEN DISCH_TAC
THEN REMOVE_THEN "DICH CHUYEN"(fun th->MP_TAC(th) THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN REMOVE_ASSUM_TAC
 THEN ASSUME_TAC(th))
THEN STRIP_TAC 
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`w' = t1'' % x + t2'' % v + t3'' % va<=>
 ((t1''+t2''+t3'')- &1) % x + t2'' % (v-x) + t3'' % (va-x)= w' -x:real^3`;REAL_ARITH`&1- &1 = &0`]
THEN REDUCE_VECTOR_TAC
THEN MP_TAC(REAL_ARITH`&0< t2'' /\ &0< t3'' /\ t1''+t2''+t3''= &1==>  &0 < &1- t1'' /\ ~(&1- t1'' = &0)/\ t2''+t3''= &1- t1''`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ISPEC`&1- (t1'':real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`&1- (t1'':real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2'' + t3'' = &1 - t1'':real
==> (inv ( &1 - t1''))*(t2'' + t3'') = (inv ( &1 - t1''))*( &1 - t1'':real)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C)= &1<=> &1- A*C = A*B`]
THEN STRIP_TAC
THEN MRESA_TAC REAL_LT_MUL[`inv(&1- t1'':real)`;`t3'':real`]
THEN MP_TAC(REAL_ARITH`&0<inv(&1- t1'':real)*(t3'':real)==> ~(inv(&1- t1'':real)*(t3'':real)= &0)`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0<inv(&1- t1'':real)==> ~(inv(&1- t1'':real)= &0)`)
THEN RESA_TAC
THEN ASSUME_TAC(REAL_ARITH`~(&1= &0)`)
THEN STRIP_TAC
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC continuous_coplanar_fan[`x:real^3 `;` v:real^3`;` u:real^3`;` w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `a:real`)
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v, (&1 - a) % u + a % w}= {(--x) +x, (--x) +v, (--x) +u, (--x) +(&1 - a) % u + a % w}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, (&1 - a) % u + a % w}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)] )
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN DISCH_TAC
THEN MRESA_TAC continuous_coplanar_fan[`(vec 0):real^3 `;` u-x:real^3`;` v-x:real^3`;` ((&1 - a) % u + a % w)-x:real^3`]
THEN POP_ASSUM (fun th-> MRESAL1_TAC th`(inv(&1- t1'')) * t3'':real`[VECTOR_ARITH`(A*B)%X+(A*C)%Y=A%(B%X+C%Y)`])
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC COPLANAR_SCALE_ALL [`&1`;`&1`;`inv(&1- t1'')`;`u - x:real^3`; `v - x:real^3`; `w'-x:real^3`]
[VECTOR_ARITH`&1 %A=A`]
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v,w'}= {(--x) +x, (--x) +u, (--x) +v, (--x) + w'}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, w'}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,C,B}`]
THEN DISCH_TAC
THEN REMOVE_THEN "EM"(fun th-> MRESA1_TAC th `w':real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN MRESA_TAC exists_element_in_half_sapace_fan
[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`w':real^3`;`  v:real^3`;`u:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u':real^3)`;
` (w':real^3)`]
THEN MRESA_TAC exists_cut_small_edges_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`;`w':real^3`;`u':real^3`]
THEN REWRITE_TAC[EMPTY;IN]
THEN MP_TAC(REAL_ARITH`&0<t /\ t< &1==> &0< &1-t:real /\ &1-t< &1`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`(&1-t:real)`;`a:real`]
THEN MRESAL_TAC REAL_LT_RMUL[`(&1-t:real)`;`&1`;`a:real`][REAL_ARITH`&1 *a=a`]
THEN REMOVE_THEN "YEU"(fun th-> MRESAL1_TAC th `(&1- t) *a`[VECTOR_ARITH`(&1 - (&1 - t) * a) % u + ((&1 - t) * a) % w=(&1 - t) % ((&1-a)%u+a % w)+t%u`;])
THEN MRESA_TAC aff_gt_subset_aff_ge[`x:real^3`;`w':real^3`;`u':real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`{w',u'} IN (E:(real^3->bool)->bool)`
THEN SET_TAC[];(*14*)

DISJ_CASES_TAC(SET_RULE`v' IN aff_gt {x} {v,va:real^3}\/ ~(v' IN aff_gt {x} {v,va:real^3})`)
THENL(*15*)[

MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`(&1-a)%u +a%w:real^3`][IN_ELIM_THM]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"DICH CHUYEN")
THEN DISCH_TAC
THEN REMOVE_THEN "DICH CHUYEN"(fun th->MP_TAC(th) THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN REMOVE_ASSUM_TAC
 THEN ASSUME_TAC(th))
THEN STRIP_TAC 
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`v' = t1'' % x + t2'' % v + t3'' % va<=>
 ((t1''+t2''+t3'')- &1) % x + t2'' % (v-x) + t3'' % (va-x)= v' -x:real^3`;REAL_ARITH`&1- &1 = &0`]
THEN REDUCE_VECTOR_TAC
THEN MP_TAC(REAL_ARITH`&0< t2'' /\ &0< t3'' /\ t1''+t2''+t3''= &1==>  &0 < &1- t1'' /\ ~(&1- t1'' = &0)/\ t2''+t3''= &1- t1''`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ISPEC`&1- (t1'':real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`&1- (t1'':real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2'' + t3'' = &1 - t1'':real
==> (inv ( &1 - t1''))*(t2'' + t3'') = (inv ( &1 - t1''))*( &1 - t1'':real)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C)= &1<=> &1- A*C = A*B`]
THEN STRIP_TAC
THEN MRESA_TAC REAL_LT_MUL[`inv(&1- t1'':real)`;`t3'':real`]
THEN MP_TAC(REAL_ARITH`&0<inv(&1- t1'':real)*(t3'':real)==> ~(inv(&1- t1'':real)*(t3'':real)= &0)`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0<inv(&1- t1'':real)==> ~(inv(&1- t1'':real)= &0)`)
THEN RESA_TAC
THEN ASSUME_TAC(REAL_ARITH`~(&1= &0)`)
THEN STRIP_TAC
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC continuous_coplanar_fan[`x:real^3 `;` v:real^3`;` u:real^3`;` w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `a:real`)
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v, (&1 - a) % u + a % w}= {(--x) +x, (--x) +v, (--x) +u, (--x) +(&1 - a) % u + a % w}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, (&1 - a) % u + a % w}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)] )
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN DISCH_TAC
THEN MRESA_TAC continuous_coplanar_fan[`(vec 0):real^3 `;` u-x:real^3`;` v-x:real^3`;` ((&1 - a) % u + a % w)-x:real^3`]
THEN POP_ASSUM (fun th-> MRESAL1_TAC th`(inv(&1- t1'')) * t3'':real`[VECTOR_ARITH`(A*B)%X+(A*C)%Y=A%(B%X+C%Y)`])
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC COPLANAR_SCALE_ALL [`&1`;`&1`;`inv(&1- t1'')`;`u - x:real^3`; `v - x:real^3`; `v'-x:real^3`]
[VECTOR_ARITH`&1 %A=A`]
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v,v'}= {(--x) +x, (--x) +u, (--x) +v, (--x) + v'}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, v'}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,C,B}`]
THEN DISCH_TAC
THEN REMOVE_THEN "EM"(fun th-> MRESA1_TAC th `v':real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN MRESA_TAC exists_element_in_half_sapace_fan
[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v':real^3`;`  v:real^3`;`u:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u':real^3)`;
` (v':real^3)`]
THEN MRESA_TAC exists_cut_small_edges_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`;`v':real^3`;`u':real^3`]
THEN REWRITE_TAC[EMPTY;IN]
THEN MP_TAC(REAL_ARITH`&0<t /\ t< &1==> &0< &1-t:real /\ &1-t< &1`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`(&1-t:real)`;`a:real`]
THEN MRESAL_TAC REAL_LT_RMUL[`(&1-t:real)`;`&1`;`a:real`][REAL_ARITH`&1 *a=a`]
THEN REMOVE_THEN "YEU"(fun th-> MRESAL1_TAC th `(&1- t) *a`[VECTOR_ARITH`(&1 - (&1 - t) * a) % u + ((&1 - t) * a) % w=(&1 - t) % ((&1-a)%u+a % w)+t%u`;])
THEN MRESA_TAC aff_gt_subset_aff_ge[`x:real^3`;`v':real^3`;`u':real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`{v',u'} IN (E:(real^3->bool)->bool)`
THEN SET_TAC[];(*15*)

SUBGOAL_THEN`~(v' IN aff_gt {x} { x',v:real^3})` ASSUME_TAC
THENL(*16*)[
REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN MRESA_TAC aff_gt3_subset_aff_gt[`x:real^3`;`v:real^3`;`va:real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC 
THEN SET_TAC[];(*16*)

MP_TAC(ISPECL[`x:real^3`;`x':real^3`;`v:real^3`;`v':real^3`] decomposition_planar_by_angle_fan)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*17*)[

MRESA_TAC aff_gt12_subset_aff_ge[`x:real^3`;`v':real^3`;`w':real^3`;`x':real^3`]
THEN MP_TAC(SET_RULE`v IN aff_gt {x} {x', v':real^3} /\ aff_gt {x} {x', v'} SUBSET aff_ge {x} {v', w'}
==>v IN aff_ge {x} {v', w':real^3}`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`; `v:real^3`; `u:real^3`; `v':real^3`;`w':real^3`]properties_of_fan7)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*18*)[
 MRESA_TAC aff_gt2_subset_aff_ge [`x:real^3`;`v:real^3`; `va:real^3`;`x':real^3`]
THEN ASSUME_TAC(REAL_ARITH`pi <= pi`)
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`x':real^3`;`v:real^3`;`va:real^3`;`w':real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`A=A+B <=> B= &0`]
THEN DISCH_TAC
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`x':real^3`;`va:real^3`;`w':real^3`]
THEN MP_TAC(ISPECL[`x:real^3`;`x':real^3`;`w':real^3`;`va:real^3`] decomposition_planar_by_angle_fan)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*19*)[

MRESA_TAC aff_gt1_subset_aff_gt[`x:real^3`;`v:real^3`; `va:real^3`;`x':real^3`]
THEN ASM_TAC THEN SET_TAC[];(*19*)

MRESA_TAC aff_gt1_subset_aff_gt[`x:real^3`;`v:real^3`; `va:real^3`;`x':real^3`]
THEN ASM_TAC THEN SET_TAC[]](*19*);(*18*)


REMOVE_THEN "CUOI" MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN REWRITE_TAC[AZIM_REFL]
THEN MP_TAC(PI_WORKS)
THEN REAL_ARITH_TAC](*18*);(*17*)

POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN DISCH_TAC
THEN MRESA_TAC aff_ge_subset_aff_gt_union_aff_ge[`(x:real^3)`;`(v:real^3)`;`(x':real^3)`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[]
THEN DISCH_TAC
THEN MP_TAC(SET_RULE`aff_ge {x} {v, x':real^3} SUBSET aff_gt {x, v} {x'} UNION aff_ge {x} {v} /\ v' IN aff_ge {x} {v, x'} ==> v' IN aff_gt {x, v} {x'} UNION aff_ge {x} {v}`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC aff_gt_inter_aff_gt[`(x:real^3)`;`(x':real^3)`;`(v:real^3)`]
THEN ASM_REWRITE_TAC[INTER;IN_ELIM_THM;UNION]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MP_TAC(ISPECL[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`; `v':real^3`; `w':real^3`; `v:real^3`;`u:real^3`]properties1_of_fan7)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> MP_TAC(SYM(th)))
THEN DISCH_TAC
THEN MRESA_TAC point_in_aff_ge[`(x:real^3)`;`(v':real^3)`;`(w':real^3)`]
THEN MRESA_TAC aff_gt2_subset_aff_ge [`x:real^3`;`v:real^3`; `va:real^3`;`x':real^3`]
THEN ASSUME_TAC(REAL_ARITH`pi <= pi`)
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`x':real^3`;`v:real^3`;`va:real^3`;`w':real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`A=A+B <=> B= &0`]
THEN DISCH_TAC
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`x':real^3`;`va:real^3`;`w':real^3`]
THEN MP_TAC(ISPECL[`x:real^3`;`x':real^3`;`w':real^3`;`va:real^3`] decomposition_planar_by_angle_fan)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*18*)[
MRESA_TAC aff_gt1_subset_aff_gt[`x:real^3`;`v:real^3`; `va:real^3`;`x':real^3`]
THEN ASM_TAC THEN SET_TAC[];(*18*)


MRESA_TAC aff_ge1_subset_aff_ge[`x:real^3`;`v':real^3`; `w':real^3`;`x':real^3`]
THEN MP_TAC(SET_RULE`va IN aff_ge {x} { x',w':real^3} /\ aff_ge {x} { x',w'} SUBSET aff_ge {x} {v', w'}
==>va IN aff_ge {x} {v', w':real^3}`)
THEN RESA_TAC
THEN SUBGOAL_THEN `va IN aff_ge {x} {u,w:real^3}` ASSUME_TAC
THENL(*19*)[

MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`u:real^3`]
THEN MRESAL_TAC AFF_GE_1_2[`x:real^3`;`u:real^3`;`w:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1 -a:real`
THEN EXISTS_TAC`a:real`
THEN MP_TAC(REAL_ARITH`&0< a /\ a  < &1 ==> &0 <= &1 - a /\ &0 <= a`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC;(*19*)

MP_TAC(SET_RULE`va IN aff_ge {x} {v', w':real^3} /\ va IN aff_ge {x} {u, w} 
==>va IN  aff_ge {x} {u, w} INTER aff_ge {x} {v', w':real^3} `)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC `FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN;fan7]
THEN STRIP_TAC
THEN POP_ASSUM  (fun th-> MP_TAC(ISPECL[`{(u:real^3),(w:real^3)}`;`{(v':real^3),(w':real^3)}`]th)) 
  THEN ASM_REWRITE_TAC[UNION;IN_ELIM_THM;]
THEN DISCH_TAC THEN ASM_REWRITE_TAC[]
THEN DISJ_CASES_TAC(SET_RULE`{u, w} INTER {v', w'}={u,w:real^3}\/ ~({u, w} INTER {v', w'}={u,w:real^3})`)
THENL(*20*)[
DISCH_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`u:real^3`]
THEN REMOVE_ASSUM_TAC
THEN MP_TAC(SET_RULE`{u, w} INTER {v', w'}={u,w:real^3}/\ ~(w=u) /\ ~(v'=w') ==> {v', w'}={u,w} `)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC `v' IN aff_ge {x} {v', w':real^3}`
THEN POP_ASSUM(fun th->REWRITE_TAC[th])
THEN MRESAL_TAC aff_ge_inter_aff_ge[`x:real^3`;`u:real^3`;`w:real^3`][INTER;IN_ELIM_THM]
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC `~collinear {x, v, u:real^3}`
THEN FIND_ASSUM MP_TAC `v=v':real^3`
THEN DISCH_TAC
THEN POP_ASSUM(fun th->REWRITE_TAC[th])
THEN DISCH_TAC
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`u:real^3`;`v:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[]
THEN DISCH_TAC
THEN MRESA_TAC AZIM_EQ_0_PI_IMP_COPLANAR[`x:real^3`;`u:real^3`;`v:real^3`;`w:real^3`]
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];(*20*)

MP_TAC(SET_RULE`~({u, w} INTER {v', w'} = {u, w:real^3}) ==> {u, w} INTER {v', w'} PSUBSET {u, w} `)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`{u, w} INTER {v', w'} PSUBSET {u, w:real^3} ==> {u, w} INTER {v', w'} SUBSET {u}\/  {u, w} INTER {v', w'} SUBSET {w} `)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*21*)[

MRESAL_TAC AFF_GE_MONO_RIGHT [`{x:real^3}`;`{u, w} INTER {v', w':real^3}`; `{u:real^3}`][DISJOINT;SET_RULE`{x} INTER {u} = {}<=> ~(x=u:real^3)`]
THEN STRIP_TAC
THEN MRESAL_TAC AFF_GE_SUBSET_AFFINE_HULL[`{x:real^3}`;`{u:real^3}`][SET_RULE`{x} UNION {u}={x,u}`;GSYM aff]
THEN MP_TAC(SET_RULE`aff_ge {x:real^3} ({u, w} INTER {v', w'}) SUBSET aff_ge {x} {u}/\ aff_ge {x} {u} SUBSET aff {x, u}/\ va IN aff_ge {x} ({u, w} INTER {v', w'})==> va IN aff {x,u}`)
THEN RESA_TAC
THEN REWRITE_TAC[IN;EMPTY]
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`u:real^3`]
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC"va"
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - a) % u + a % w = u' % x + v'' % u
<=> a % w = u' % x + (v''+a- &1) % u`]
THEN MRESA1_TAC REAL_MUL_LINV`a:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`a % w = u' % x + (v'' + a - &1) % u:real^3 ==> (inv (a))%(a % w ) = (inv (a))%( u' % x + (v'' + a - &1) % u)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`(inv a * u'):real`
THEN EXISTS_TAC`(inv a * (v'' + a - &1)):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv a * u' + inv a * (v'' + a - &1) =inv a * ( a+ (u' + v'') - &1)`;REAL_ARITH`A+ &1- &1=A`];(*21*)


STRIP_TAC
THEN FIND_ASSUM MP_TAC`{u,w:real^3} IN E`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN STRIP_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` u:real^3`;`w:real^3`]
THEN MRESAL_TAC AFF_GE_MONO_RIGHT [`{x:real^3}`;`{u, w} INTER {v', w':real^3}`; `{w:real^3}`][DISJOINT;SET_RULE`{x} INTER {w} = {}<=> ~(x=w:real^3)`]
THEN MRESAL_TAC AFF_GE_SUBSET_AFFINE_HULL[`{x:real^3}`;`{w:real^3}`][SET_RULE`{x} UNION {u}={x,u}`;GSYM aff]
THEN MP_TAC(SET_RULE`aff_ge {x:real^3} ({u, w} INTER {v', w'}) SUBSET aff_ge {x} {w}/\ aff_ge {x} {w} SUBSET aff {x, w}/\ va IN aff_ge {x} ({u, w} INTER {v', w'})==> va IN aff {x,w}`)
THEN RESA_TAC
THEN REWRITE_TAC[IN;EMPTY]
THEN POP_ASSUM MP_TAC 
THEN FIND_ASSUM MP_TAC `~(u IN aff {x, w:real^3})`
THEN REWRITE_TAC[]
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC"va"
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - a) % u + a % w = u' % x + v'' % w
<=> (&1-a )% u = u' % x + (v''-a) % w`]
THEN MP_TAC(REAL_ARITH`a< &1==> ~(&1-a= &0)` )THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`&1-a:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(&1-a )% u = u' % x + (v''-a) % w:real^3 ==> (inv (&1-a))%((&1-a )% u ) = (inv (&1-a))%( u' % x + (v''-a) % w)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`(inv (&1-a) * u'):real`
THEN EXISTS_TAC`(inv (&1-a) * (v'' - a)):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (&1 - a) * u' + inv (&1 - a) * (v'' - a)=inv (&1 - a) * ((u' + v'') - a)`]](*21*)](*20*)](*19*)](*18*)](*17*)](*16*)](*15*)](*14*)](*13*)](*12*)](*11*)](*10*)](*9*)]
(*8*)](*7*)](*6*)](*5*)](*4*)](*3*)](*2*);(*1*)

SET_TAC[]]);;



let aff_ge_eq_aff_gt_union_aff_ge=prove(`!(x:real^3) (v:real^3) (w:real^3).
~collinear {x,v,w}
==>
aff_ge {x} {v , w} =  (aff_gt {x} {v,w}) UNION (aff_ge {x} {v}) UNION (aff_ge {x} {w})`,

REPEAT STRIP_TAC THEN MRESA_TAC th3 [`x:real^3`;`v:real^3`;`w:real^3`]
  THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`]AFF_GE_1_2)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
		THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`]AFF_GT_1_2)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
		THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`]AFF_GE_1_1)
		THEN MP_TAC(ISPECL[`x:real^3`;`w:real^3`]AFF_GE_1_1) THEN RESA_TAC THEN RESA_TAC
		THEN ASM_REWRITE_TAC[EXTENSION;UNION;IN_ELIM_THM]
THEN GEN_TAC THEN EQ_TAC
THENL[STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0<= t2/\ &0<=t3==> (t2= &0)\/ (t3= &0)\/ (&0<t2 /\ &0<t3)`)
THEN RESA_TAC
THENL[
SUBGOAL_THEN `(?t1' t2'.
      &0 <= t2' /\
      t1' + t2' = &1 /\
      t1 % x + &0 % v + t3 % w = t1' % x + t2' % w:real^3)`ASSUME_TAC
THENL[
EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t3:real`
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC;
POP_ASSUM MP_TAC THEN SET_TAC[]];

SUBGOAL_THEN `(?t1' t2'.
      &0 <= t2' /\
      t1' + t2' = &1 /\
      t1 % x + t2 % v + &0 % w = t1' % x + t2' % v:real^3)`ASSUME_TAC
THENL[
EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC;
POP_ASSUM MP_TAC THEN SET_TAC[]];

SUBGOAL_THEN `(?t1' t2' t3'.
      &0 < t2' /\
      &0 < t3' /\
      t1' + t2' + t3' = &1 /\
      t1 % x + t2 % v + t3 % w = t1' % x + t2' % v + t3' % w:real^3)`ASSUME_TAC
THENL[
EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3:real`
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC;
POP_ASSUM MP_TAC THEN SET_TAC[]]];

STRIP_TAC
THENL[
EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3:real`
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC;
EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`&0:real`
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC;
EXISTS_TAC`t1:real`
THEN EXISTS_TAC`&0:real`
THEN EXISTS_TAC`t2:real`
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC]]);;
let AFFINE_HULL_1=prove(`!a. affine hull {a} ={u % a| u = &1}`, 
  SIMP_TAC[AFFINE_HULL_FINITE; FINITE_INSERT; FINITE_RULES] THEN
  SIMP_TAC[AFFINE_HULL_FINITE_STEP; FINITE_INSERT; FINITE_RULES] THEN
  REWRITE_TAC[REAL_ARITH `x - y = z:real <=> x = y + z`;
              VECTOR_ARITH `x - y = z:real^N <=> x = y + z`] THEN
  REWRITE_TAC[VECTOR_ADD_RID; REAL_ADD_RID] THEN SET_TAC[]);;

let aff_ge1_1_subset_aff_ge=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
DISJOINT {x} {v,u} /\ ~(x=v1)
/\ v1 IN aff_ge {x} {v,u}
==>
aff_ge {x} {v1} SUBSET aff_ge {x} {v,u}`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GE_1_1[`x:real^3`;`v1:real^3`;][IN_ELIM_THM;SUBSET]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % (t1 % x + t2 % v + t3 % u) 
=(t1'+ t2'*t1) % x + (t2'* t2) % v + (t2' * t3 ) % u:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`t1' + t2' * t1:real`
THEN EXISTS_TAC`t2' * t2:real`
THEN EXISTS_TAC`t2' * t3:real`
THEN MP_TAC(REAL_ARITH`&0<t2'==> &0<= t2'`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t2:real`]
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t3:real`]
THEN ASM_REWRITE_TAC[REAL_ARITH`(t1' + t2' * t1) + t2' * t2 + t2' * t3 =
t1'+t2'*(t1+t2+t3)`; REAL_ARITH`A* &1=A`]
THEN ASM_TAC
THEN REAL_ARITH_TAC);;


let exist_close1_fan=prove(`(!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) (w:real^3) (v1:real^3) (w1:real^3) a:real.

FAN(x,V,E) /\ {v,u} INTER {v1,w1}={} /\ {v1,w1} IN E /\ {v,u} IN E /\ {u,w} IN E
/\ (&0< azim x u w v ) /\ (azim x u w v <pi)
/\ &0 <a /\ a< &1 
/\(aff_gt {x} {v, (&1 - a) % u + a % w} INTER
      {v | ?e. e IN E /\ v IN aff_ge {x} e} =     {})
==>
?h:real.
(&0 < h) 
/\ 
(!y1:real^3 y2:real^3. y1 IN (aff_ge {x} {v, (&1 - a) % u + a % w} INTER ballnorm_fan x) /\ y2 IN (aff_ge {x} {v1, w1} INTER ballnorm_fan x)
==> h  <= dist(y1,y2) ))`,
REPEAT STRIP_TAC 
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"EM")
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w1:real^3)`;
` (v1:real^3)`]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`u:real^3`]
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded[`(x:real^3)` ;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (v:real^3)`;`(u:real^3)`;`(w:real^3)`;`a:real`]

THEN MATCH_MP_TAC( ISPECL [`aff_ge {(x:real^3)} {(v:real^3), ((&1 - a) % u + a % w:real^3)} INTER ballnorm_fan x`; 
 `aff_ge {(x:real^3)} {(v1:real^3), (w1:real^3)} INTER ballnorm_fan x`] SEPARATE_CLOSED_COMPACT)
THEN MP_TAC(ISPECL[`(x:real^3) `;` (v:real^3)`;` ((&1 - a) % u + a % w:real^3)`]closed_aff_ge_ballnorm_fan) 
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(x:real^3) `;` (v1:real^3)`;` (w1:real^3)`]compact_aff_ge_ballnorm_fan) THEN RESA_TAC
THEN REMOVE_THEN "EM" MP_TAC
THEN FIND_ASSUM MP_TAC `{v1,w1:real^3} IN E`
THEN MRESA_TAC aff_ge_eq_aff_gt_union_aff_ge[`(x:real^3)`;` (v:real^3)`;`((&1 - a) % u + a % w:real^3)`]
THEN FIND_ASSUM(MP_TAC)`FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN;fan7] 
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MP_TAC(ISPECL[`{(v:real^3)}`;`{(v1:real^3),(w1:real^3)}`]th))
		   THEN ASM_REWRITE_TAC[UNION; IN_ELIM_THM;]
THEN MP_TAC(SET_RULE`(v:real^3) IN V==>(?v'. v' IN V /\ {v} = {v'})`)
THEN REDA_TAC
THEN MP_TAC(SET_RULE`{(v:real^3),u} INTER {v1,w1}={} ==> {v} INTER {v1,w1}={}`)
THEN REDA_TAC
THEN REWRITE_TAC[AFF_GE_EQ_AFFINE_HULL;AFFINE_HULL_1;]
THEN STRIP_TAC THEN SUBGOAL_THEN`(aff_ge {x} {v} INTER aff_ge {x} {v1, w1:real^3}) INTER ballnorm_fan x={}`
ASSUME_TAC
THENL(*1*)[
ASM_REWRITE_TAC[INTER;ballnorm_fan;EXTENSION;IN_ELIM_THM;SET_RULE`(?u. u = &1 /\ x' = u % x)<=> x'= &1 % x`;]
THEN REDUCE_VECTOR_TAC
THEN GEN_TAC THEN EQ_TAC
THENL(*2*)[
STRIP_TAC THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[DIST_REFL]
THEN REAL_ARITH_TAC;
SET_TAC[]];

ABBREV_TAC`va=(&1 - a) % u + a % w:real^3`
 THEN SUBGOAL_THEN `va IN aff_ge {x} {u,w:real^3}` ASSUME_TAC
THENL(*2*)[
 MRESAL_TAC AFF_GE_1_2[`x:real^3`;`u:real^3`;`w:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1 -a:real`
THEN EXISTS_TAC`a:real`
THEN MP_TAC(REAL_ARITH`&0< a /\ a  < &1 ==> &0 <= &1 - a /\ &0 <= a`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC;(*2*)

SUBGOAL_THEN`(aff_ge {x} {va} INTER aff_ge {x} {v1, w1:real^3}) SUBSET {x}`
ASSUME_TAC
THENL(*3*)[
 MRESA_TAC th3[` (x:real^3)`;`(v:real^3)`;`(va:real^3)`]
THEN MRESA_TAC aff_ge1_1_subset_aff_ge[`x:real^3`;`u:real^3`;`w:real^3`;`va:real^3`]
THEN FIND_ASSUM(MP_TAC)`FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN;fan7] 
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MP_TAC(ISPECL[`{(u:real^3),(w:real^3)}`;`{(v1:real^3),(w1:real^3)}`]th))
		   THEN ASM_REWRITE_TAC[UNION; IN_ELIM_THM;]
THEN MP_TAC(SET_RULE`{v,u} INTER {v1,w1}={}==> {u,w} INTER {v1,w1} SUBSET {w:real^3}`)
THEN RESA_TAC
THEN STRIP_TAC
THEN MRESAL_TAC AFF_GE_MONO_RIGHT [`{x:real^3}`;`{u, w} INTER {v1, w1:real^3}`; `{w:real^3}`][DISJOINT;SET_RULE`{x} INTER {u} = {}<=> ~(x=u:real^3)`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th)])
THEN REMOVE_ASSUM_TAC
THEN DISCH_TAC
THEN MP_TAC(SET_RULE`aff_ge {x} {va} SUBSET aff_ge {x} {u, w}/\
aff_ge {x} {u, w} INTER aff_ge {x} {v1, w1} SUBSET aff_ge {x} {w}
==> aff_ge {x} {va:real^3} INTER aff_ge {x} {v1, w1} SUBSET  aff_ge {x} {va} INTER aff_ge {x} {w}`)
THEN RESA_TAC
THEN MATCH_MP_TAC(SET_RULE` aff_ge {x} {va:real^3} INTER aff_ge {x} {v1, w1} SUBSET  aff_ge {x} {va} INTER aff_ge {x} {w} /\ aff_ge {x} {va} INTER aff_ge {x} {w} SUBSET {x:real^3} ==>aff_ge {x} {va:real^3} INTER aff_ge {x} {v1, w1} SUBSET {x}`)
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC  AFF_GE_1_1[`x:real^3`;`va:real^3`;][IN_ELIM_THM;SUBSET;INTER]
THEN MRESAL_TAC  AFF_GE_1_1[`x:real^3`;`w:real^3`;][IN_ELIM_THM;SUBSET;IN_SING]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN EXPAND_TAC"va"
THEN REWRITE_TAC[VECTOR_ARITH`t1 % x + t2 % ((&1 - a) % u + a % w) = t1' % x + t2' % w
<=> (t2 *(&1 - a)) % u = (t1'-t1) % x + (t2'- t2*a) % w:real^3`]
THEN MP_TAC(REAL_ARITH`&0<= t2==> t2= &0 \/ &0 < t2:real`)
THEN RES_TAC
THENL(*4*)[
FIND_ASSUM MP_TAC`t1+ t2:real= &1`
THEN POP_ASSUM (fun th-> REWRITE_TAC[th])
THEN REDUCE_ARITH_TAC
THEN DISCH_TAC
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC;(*4*)

MP_TAC(REAL_ARITH`a< &1==> &0< &1-a:real`) THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`t2:real`;`&1- a:real`]
THEN MP_TAC(REAL_ARITH` &0< t2*(&1-a)==> ~(t2*(&1-a):real= &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`t2*(&1-a):real`
THEN MRESA1_TAC REAL_LT_INV`t2*(&1-a):real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(t2 * (&1 - a)) % u = (t1' - t1) % x + (t2' - t2 * a) % w:real^3 ==> (inv (t2 * (&1 - a)))%((t2 * (&1 - a)) % u ) = (inv (t2 * (&1 - a)))%((t1' - t1) % x + (t2' - t2 * a) % w)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN`u IN aff {x,w:real^3}`ASSUME_TAC
THENL(*5*)[
REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN EXISTS_TAC`(inv (t2 * (&1 - a)) * (t1' - t1)):real`
THEN EXISTS_TAC`(inv (t2 * (&1 - a)) * (t2' - t2 * a)):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (t2 * (&1 - a)) * (t1' - t1) + inv (t2 * (&1 - a)) * (t2' - t2 * a)
= inv (t2 * (&1 - a)) * ((t1' +t2')- (t1+t2)+ t2*(&1 - a)):real`; REAL_ARITH`&1- &1 + A= A`];(*5*)

FIND_ASSUM MP_TAC `~collinear {x, u, w:real^3}`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN STRIP_TAC
THEN MRESA_TAC th3[`x:real^3`;`w:real^3`;`u:real^3`]]];(*3*)


SUBGOAL_THEN`(aff_ge {x} {va} INTER aff_ge {x} {v1, w1:real^3}) INTER ballnorm_fan x={}`
ASSUME_TAC
THENL(*4*)[
ASM_REWRITE_TAC[INTER;ballnorm_fan;EXTENSION;IN_ELIM_THM;]
THEN REDUCE_VECTOR_TAC
THEN GEN_TAC THEN EQ_TAC
THENL(*5*)[
STRIP_TAC THEN POP_ASSUM MP_TAC
THEN MP_TAC(SET_RULE`x' IN aff_ge {x} {va} /\ x' IN aff_ge {x} {v1, w1}
/\ aff_ge {x} {va} INTER aff_ge {x} {v1, w1} SUBSET {x}
==> x' IN {x:real^3}`)
THEN REWRITE_TAC[IN_SING]
THEN RESA_TAC
THEN ASM_REWRITE_TAC[DIST_REFL;]
THEN REAL_ARITH_TAC;(*5*)
SET_TAC[]];
ASM_TAC THEN SET_TAC[]]]]]);;

let properties_inside_collinear0_fan=prove(`!(x:real^3)  (u:real^3) (w:real^3) a:real.
&0 <a /\ a< &1 
/\ ~collinear{x,w,u}
==> ~collinear{x,(&1 - a) % u + a % w,u}`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[collinear1_fan]
THEN STRIP_TAC THEN ASM_REWRITE_TAC[]
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC 
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff; AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - a) % u + a % w = u' % x + v % u
<=> a % w = u' % x + (v+a- &1) % u`]
THEN MP_TAC(REAL_ARITH`&0< a ==> ~(a= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`a:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
a % w = u' % x + (v+a- &1) % u:real^3
==> (inv ( a))%(a % w) = (inv (a))%(u' % x + (v+a- &1) % u)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C= (A*B)%C`;VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC `(inv a * (u':real))`
THEN EXISTS_TAC `(inv a * (v +a - &1 :real))`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv a * (u') + inv a * (v +a - &1)=inv a* (a+ (u'+v) - &1)`;REAL_ARITH`t3'+ &1 - &1=t3'`]);;

let properties_inside_collinear1_fan=prove(`!(x:real^3)  (u:real^3) (w:real^3) a:real.
&0 <a /\ a< &1 
/\ ~collinear{x,w,u}
==>  ~collinear{x,(&1 - a) % u + a % w,w}`,
REPEAT STRIP_TAC THEN
MRESAL_TAC properties_inside_collinear0_fan[`(x:real^3)`;` (w:real^3)`;`(u:real^3)`;`&1-a:real`][VECTOR_ARITH`(&1 - (&1 - a)) % w + (&1 - a) % u=(&1 - a) % u + a % w`;]
THENL[ ASM_TAC THEN REAL_ARITH_TAC;
STRIP_TAC THENL[ASM_TAC THEN REAL_ARITH_TAC;
ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[]]]);;
let properties1_inside_fan=prove(`!x:real^3 u:real^3 w:real^3.
DISJOINT {x} {u,w}
/\ &0<a /\ a< &1
==> (&1-a)%u+ a%w IN aff_ge {x} {u,w:real^3}`,
REPEAT STRIP_TAC
THEN  MRESAL_TAC AFF_GE_1_2[`x:real^3`;`u:real^3`;`w:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1 -a:real`
THEN EXISTS_TAC`a:real`
THEN MP_TAC(REAL_ARITH`&0< a /\ a  < &1 ==> &0 <= &1 - a /\ &0 <= a`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC);;


let properties_inside_collinear_fan=prove(`!(x:real^3)  (u:real^3) (w:real^3) a:real.
&0 <a /\ a< &1 
/\ ~collinear{x,u,w}
==>  ~collinear{x,(&1 - a) % u + a % w,u}
/\  ~collinear{x,(&1 - a) % u + a % w,w}`,

MESON_TAC[SET_RULE`{A,B,C}={A,C,B}`;properties_inside_collinear0_fan;properties_inside_collinear1_fan]
);;


let properties_inside_collinear1_fan=prove(`!x:real^3 u:real^3 w:real^3.
~collinear{x,u,w}
/\ &0<a /\ a< &1
==>  aff_ge {x} {u} INTER aff_ge {x} {(&1-a)%u+ a%w,w:real^3} SUBSET aff_ge {x} {}`,

REPEAT STRIP_TAC
THEN MRESA_TAC properties_inside_collinear_fan[`(x:real^3)`;`(u:real^3)`;`(w:real^3)`;`a:real`]
THEN MRESA_TAC th3[`(x:real^3)`;`((&1-a)%u+ a%w:real^3)`;`(w:real^3)`]
THEN MRESA_TAC th3[`(x:real^3)`;`u:real^3`;`(w:real^3)`]
THEN  MRESAL_TAC AFF_GE_1_2[`x:real^3`;`(&1-a)%u+ a%w:real^3`;`w:real^3`][IN_ELIM_THM]
THEN  MRESAL_TAC AFF_GE_1_1[`x:real^3`;`u:real^3`][IN_ELIM_THM;INTER;SUBSET;AFF_GE_EQ_AFFINE_HULL;AFFINE_HULL_1]
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`&1`
THEN REDUCE_VECTOR_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN MP_TAC(REAL_ARITH`&0<= t2' /\ &0 <= t3==> (t2'= &0 /\ t3 = &0)\/ (&0< t2' \/ &0 <t3)  `)
THEN RESA_TAC
THENL[REDUCE_ARITH_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC;
STRIP_TAC THEN 
REWRITE_TAC[VECTOR_ARITH`t1 % x + t2 % u = t1' % x + t2' % ((&1 - a) % u + a % w) + t3 % w
<=>  (t2'* a + t3) % w = (t1-t1') % x +( t2-t2' * (&1-a))% u   :real^3`]
THEN MRESA_TAC REAL_LT_MUL[`t2':real`;`a:real`]
THEN MP_TAC(REAL_ARITH` &0< t2'*(a) /\ &0<= t3 ==> &0 < t2'*(a)+t3 /\ ~(t2'*(a)+t3:real= &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`t2'*(a)+t3:real`
THEN MRESA1_TAC REAL_LT_INV`t2'*(a)+t3:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(t2'* a + t3) % w = (t1-t1') % x +( t2-t2' * (&1-a))% u:real^3
 ==> (inv (t2' * a+t3))%((t2'* a + t3) % w) = (inv (t2' * a+t3))%( (t1-t1') % x +( t2-t2' * (&1-a))% u)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN`w IN aff {x,u:real^3}`ASSUME_TAC
THENL[REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN EXISTS_TAC`(inv (t2' * a+t3)) * (t1 - t1'):real`
THEN EXISTS_TAC`(inv (t2' *  a+t3) * (t2 - t2' *(&1- a))):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (t2' * a + t3) * (t1 - t1') + inv (t2' * a + t3) * (t2 - t2' * (&1 - a))
= inv (t2' * a + t3) * ((t2'*a +t3)+(t1 + t2) -(t1'+ t2' +t3) ):real`; REAL_ARITH`A+ &1- &1= A`];
ASM_MESON_TAC[]];

STRIP_TAC THEN 
REWRITE_TAC[VECTOR_ARITH`t1 % x + t2 % u = t1' % x + t2' % ((&1 - a) % u + a % w) + t3 % w
<=>  (t2'* a + t3) % w = (t1-t1') % x +( t2-t2' * (&1-a))% u   :real^3`]
THEN MP_TAC(REAL_ARITH`&0<a==> &0<=a`) THEN RESA_TAC
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`a:real`]
THEN MP_TAC(REAL_ARITH` &0<= t2'*(a) /\ &0< t3 ==> &0 < t2'*(a)+t3 /\ ~(t2'*(a)+t3:real= &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`t2'*(a)+t3:real`
THEN MRESA1_TAC REAL_LT_INV`t2'*(a)+t3:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(t2'* a + t3) % w = (t1-t1') % x +( t2-t2' * (&1-a))% u:real^3
 ==> (inv (t2' * a+t3))%((t2'* a + t3) % w) = (inv (t2' * a+t3))%( (t1-t1') % x +( t2-t2' * (&1-a))% u)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN`w IN aff {x,u:real^3}`ASSUME_TAC
THENL[
REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN EXISTS_TAC`(inv (t2' * a+t3)) * (t1 - t1'):real`
THEN EXISTS_TAC`(inv (t2' *  a+t3) * (t2 - t2' *(&1- a))):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (t2' * a + t3) * (t1 - t1') + inv (t2' * a + t3) * (t2 - t2' * (&1 - a))
= inv (t2' * a + t3) * ((t2'*a +t3)+(t1 + t2) -(t1'+ t2' +t3) ):real`; REAL_ARITH`A+ &1- &1= A`];
ASM_MESON_TAC[]]]);;


let lemma_proof0_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (u:real^3) (w:real^3) (v1:real^3) (w1:real^3) a:real.

FAN(x,V,E)  /\ {v1,w1} IN E /\ {u,w} IN E /\ ~(u IN {v1,w1})
/\ &0 <a /\ a< &1 
==> (aff_ge {x} {(&1 - a) % u + a % w} INTER aff_ge {x} {v1, w1:real^3}) SUBSET {x}`,
REPEAT STRIP_TAC 
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w1:real^3)`;
` (v1:real^3)`]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`u:real^3`]
THEN FIND_ASSUM MP_TAC `{v1,w1:real^3} IN E`
THEN ABBREV_TAC`va=(&1 - a) % u + a % w:real^3`
THEN MRESA_TAC properties1_inside_fan[`(x:real^3)`;` (u:real^3)`;`(w:real^3)`;]
THEN MRESA_TAC properties_inside_collinear_fan[`(x:real^3)`;` (u:real^3)`;`(w:real^3)`;`a:real`]
THEN MRESA_TAC th3[` (x:real^3)`;`(va:real^3)`;`(u:real^3)`]
THEN MRESA_TAC aff_ge1_1_subset_aff_ge[`x:real^3`;`u:real^3`;`w:real^3`;`va:real^3`]
THEN FIND_ASSUM(MP_TAC)`FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN;fan7] 
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MP_TAC(ISPECL[`{(u:real^3),(w:real^3)}`;`{(v1:real^3),(w1:real^3)}`]th))
		   THEN ASM_REWRITE_TAC[UNION; IN_ELIM_THM;]
THEN MP_TAC(SET_RULE`~(u IN {v1,w1})==> {u,w} INTER {v1,w1} SUBSET {w:real^3}`)
THEN RESA_TAC
THEN STRIP_TAC
THEN MRESAL_TAC AFF_GE_MONO_RIGHT [`{x:real^3}`;`{u, w} INTER {v1, w1:real^3}`; `{w:real^3}`][DISJOINT;SET_RULE`{x} INTER {u} = {}<=> ~(x=u:real^3)`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th)])
THEN REMOVE_ASSUM_TAC
THEN DISCH_TAC
THEN MP_TAC(SET_RULE`aff_ge {x} {va} SUBSET aff_ge {x} {u, w}/\
aff_ge {x} {u, w} INTER aff_ge {x} {v1, w1} SUBSET aff_ge {x} {w}
==> aff_ge {x} {va:real^3} INTER aff_ge {x} {v1, w1} SUBSET  aff_ge {x} {va} INTER aff_ge {x} {w}`)
THEN RESA_TAC
THEN MATCH_MP_TAC(SET_RULE` aff_ge {x} {va:real^3} INTER aff_ge {x} {v1, w1} SUBSET  aff_ge {x} {va} INTER aff_ge {x} {w} /\ aff_ge {x} {va} INTER aff_ge {x} {w} SUBSET {x:real^3} ==>aff_ge {x} {va:real^3} INTER aff_ge {x} {v1, w1} SUBSET {x}`)
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC  AFF_GE_1_1[`x:real^3`;`va:real^3`;][IN_ELIM_THM;SUBSET;INTER]
THEN MRESAL_TAC  AFF_GE_1_1[`x:real^3`;`w:real^3`;][IN_ELIM_THM;SUBSET;IN_SING]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN EXPAND_TAC"va"
THEN REWRITE_TAC[VECTOR_ARITH`t1 % x + t2 % ((&1 - a) % u + a % w) = t1' % x + t2' % w
<=> (t2 *(&1 - a)) % u = (t1'-t1) % x + (t2'- t2*a) % w:real^3`]
THEN MP_TAC(REAL_ARITH`&0<= t2==> t2= &0 \/ &0 < t2:real`)
THEN RES_TAC
THENL(*4*)[
FIND_ASSUM MP_TAC`t1+ t2:real= &1`
THEN POP_ASSUM (fun th-> REWRITE_TAC[th])
THEN REDUCE_ARITH_TAC
THEN DISCH_TAC
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC;
MP_TAC(REAL_ARITH`a< &1==> &0< &1-a:real`) THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`t2:real`;`&1- a:real`]
THEN MP_TAC(REAL_ARITH` &0< t2*(&1-a)==> ~(t2*(&1-a):real= &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`t2*(&1-a):real`
THEN MRESA1_TAC REAL_LT_INV`t2*(&1-a):real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(t2 * (&1 - a)) % u = (t1' - t1) % x + (t2' - t2 * a) % w:real^3 ==> (inv (t2 * (&1 - a)))%((t2 * (&1 - a)) % u ) = (inv (t2 * (&1 - a)))%((t1' - t1) % x + (t2' - t2 * a) % w)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN`u IN aff {x,w:real^3}`ASSUME_TAC
THENL(*5*)[
REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN EXISTS_TAC`(inv (t2 * (&1 - a)) * (t1' - t1)):real`
THEN EXISTS_TAC`(inv (t2 * (&1 - a)) * (t2' - t2 * a)):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (t2 * (&1 - a)) * (t1' - t1) + inv (t2 * (&1 - a)) * (t2' - t2 * a)
= inv (t2 * (&1 - a)) * ((t1' +t2')- (t1+t2)+ t2*(&1 - a)):real`; REAL_ARITH`&1- &1 + A= A`];
FIND_ASSUM MP_TAC `~collinear {x, u, w:real^3}`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN STRIP_TAC
THEN MRESA_TAC th3[`x:real^3`;`w:real^3`;`u:real^3`]]]);;

let lemma_proof1_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (u:real^3) (w:real^3) (v1:real^3) (w1:real^3) a:real.

FAN(x,V,E)  /\ {v1,w1} IN E /\ {u,w} IN E /\ ~(w IN {v1,w1})
/\ &0 <a /\ a< &1 
==> (aff_ge {x} {(&1 - a) % u + a % w} INTER aff_ge {x} {v1, w1:real^3}) SUBSET {x}`,

REPEAT STRIP_TAC
THEN MRESAL_TAC lemma_proof0_fan[`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(w:real^3)`;` (u:real^3)`;`(v1:real^3)`;`(w1:real^3)`;`&1 -a:real`][VECTOR_ARITH`(&1 - (&1 - a)) % w + (&1 - a) % u=(&1 - a) % u + a % w`;]
THEN POP_ASSUM MATCH_MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[] THEN ASM_TAC THEN REAL_ARITH_TAC);;


let GRAPH = prove
 (`!E. graph E <=> !e. e IN E ==> e HAS_SIZE 2`,
  REWRITE_TAC[graph; IN]);;


let CARD_2_FAN=prove(`!v:A w:A. ~(v=w) 
==> CARD {v,w}=2`,
REPEAT STRIP_TAC
THEN SUBGOAL_THEN`FINITE {v,w:A}`ASSUME_TAC
THENL[      SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY;
                 IN_INSERT; NOT_IN_EMPTY];

ASSUME_TAC(SET_RULE `v:A IN {v:A,w:A} `)
THEN MP_TAC(ISPECL[`v:A`;`{v:A,w:A}`;]CARD_DELETE)
THEN RESA_TAC
THEN MP_TAC(SET_RULE `v IN {v,w}==>{v:A,w:A} DELETE v PSUBSET {v,w}`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`{v:A,w:A} DELETE v`;`{v:A,w:A}`]CARD_PSUBSET)
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN FIND_ASSUM MP_TAC`FINITE {v:A,w:A}`
THEN DISCH_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN DISCH_TAC
THEN MP_TAC(ARITH_RULE`CARD ({v, w} DELETE v) < CARD {v, w}/\ CARD ({v, w} DELETE v) = CARD {v, w}-1
<=>CARD ({v, w} DELETE v) +1= CARD {v:A, w:A}`)
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th;])
THEN REWRITE_TAC[ARITH_RULE`A=A`]
THEN DISCH_TAC
THEN SUBGOAL_THEN `w:A IN ({v:A,w:A} DELETE v)` ASSUME_TAC
THENL[
ASM_SET_TAC[];
MP_TAC(ISPECL[`{v:A,w:A}`;`v:A`] FINITE_DELETE)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`w:A`;`{v:A,w:A} DELETE v`;]CARD_DELETE)
THEN RESA_TAC
THEN MP_TAC(SET_RULE `w IN ({v,w} DELETE v)==>{v:A,w:A} DELETE v DELETE w PSUBSET {v,w} DELETE v`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`{v:A,w:A} DELETE v DELETE w`;`{v:A,w:A} DELETE v`]CARD_PSUBSET)
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN FIND_ASSUM MP_TAC`FINITE ({v:A,w:A} DELETE v)`
THEN DISCH_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN DISCH_TAC
THEN MP_TAC(ARITH_RULE`CARD ({v, w} DELETE v DELETE w) < CARD ({v, w} DELETE v)/\ CARD ({v, w} DELETE v DELETE w) = CARD ({v, w} DELETE v)-1
<=>CARD ({v, w} DELETE v DELETE w) +1= CARD ({v:A, w:A} DELETE v)`)
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th;])
THEN REWRITE_TAC[ARITH_RULE`A=A`]
THEN DISCH_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[])
THEN POP_ASSUM (fun th->REWRITE_TAC[])
THEN ASSUME_TAC(SET_RULE `{v, w} DELETE v:A DELETE w:A={}`)
THEN POP_ASSUM (fun th->REWRITE_TAC[th;CARD_CLAUSES; ARITH_RULE `0+1=1`])
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"B")
THEN DISCH_TAC
THEN REMOVE_THEN "B" MP_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th);ARITH_RULE` 1+1=2`])
THEN SET_TAC[]]]);; 




let lemma_proof_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (u:real^3) (w:real^3) (v1:real^3) (w1:real^3) a:real.

FAN(x,V,E)  /\ {v1,w1} IN E /\ {u,w} IN E /\ ~({u,w} = {v1,w1})
/\ &0 <a /\ a< &1 
==> (aff_ge {x} {(&1 - a) % u + a % w} INTER aff_ge {x} {v1, w1:real^3}) SUBSET {x}`,

REPEAT STRIP_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w1:real^3)`;
` (v1:real^3)`]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`u:real^3`]
THEN MP_TAC(SET_RULE` ~(u=w) /\ ~(v1=w1:real^3)==> ({u,w} = {v1,w1}<=> {u,w} SUBSET {v1,w1})`)
THEN ASM_REWRITE_TAC[SET_RULE`~({u,w} SUBSET {v1,w1})<=> ~(w IN {v1,w1})\/ ~( u IN {v1,w1})`]
THEN ASM_TAC THEN SET_TAC[lemma_proof0_fan;lemma_proof1_fan]);;



let remark012_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (u:real^3) (w:real^3)  a:real.
FAN(x,V,E)  /\ {u,w} IN E 
/\ &0 <a /\ a< &1 
==> ~((&1 - a) % u + a % w IN V)`,
REPEAT STRIP_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`u:real^3`]
THEN ABBREV_TAC`v1=(&1 - a) % u + a % w:real^3`
THEN FIND_ASSUM MP_TAC `FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN;fan2] 
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(SET_RULE`~(x IN V) /\ v1 IN V==> ~(x=(v1:real^3))`) 
THEN RESA_TAC
THEN MRESA_TAC properties_inside_collinear_fan[`(x:real^3)`;`(u:real^3)`;`(w:real^3)`;`a:real`]
THEN MRESA_TAC th3[`(x:real^3)`;`((&1-a)%u+ a%w:real^3)`;`(w:real^3)`]
THEN MRESA_TAC th3[`(x:real^3)`;`((&1-a)%u+ a%w:real^3)`;`(u:real^3)`]
THEN MP_TAC(SET_RULE`DISJOINT {x, v1} {u}==> ~(v1=u:real^3)`) THEN RESA_TAC
THEN MP_TAC(SET_RULE`DISJOINT {x, v1} {w}==> ~(v1=w:real^3)`) THEN RESA_TAC
THEN MP_TAC(SET_RULE` ~(v1=w:real^3) /\ ~(v1=u:real^3) ==> {u,w} INTER {v1}={}`) THEN RESA_TAC
THEN FIND_ASSUM MP_TAC `FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN;fan7] 
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MP_TAC(ISPECL[`{(u:real^3),w}`;`({v1:real^3})`]th))
		   THEN ASM_REWRITE_TAC[UNION; IN_ELIM_THM;]
THEN MP_TAC(SET_RULE`(v1:real^3) IN V==>(?v'. v' IN V /\ {v1} = {v'})`)
THEN REDA_TAC
THEN MRESA_TAC properties1_inside_fan[`(x:real^3)`;` (u:real^3)`;`(w:real^3)`;]
THEN MRESA_TAC aff_ge1_subset_aff_ge[`x:real^3`;`u:real^3`; `w:real^3`;`v1:real^3`]
THEN MRESA_TAC aff_ge_eq_aff_gt_union_aff_ge[`(x:real^3)`;`((&1 - a) % u + a % w:real^3)`;` (w:real^3)`]
THEN MP_TAC(SET_RULE`aff_ge {x} {v1, w} SUBSET aff_ge {x} {u, w}/\ aff_ge {x} {v1, w} =
      aff_gt {x} {v1, w} UNION aff_ge {x} {v1} UNION aff_ge {x} {w}
==> aff_ge {x} {u, w} INTER aff_ge {x} {v1}=aff_ge {x} {v1:real^3}`)
THEN RESA_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN`v1 IN aff_ge {x} {v1:real^3}` ASSUME_TAC
THENL[MRESAL_TAC  AFF_GE_1_1[`x:real^3`;`v1:real^3`;][IN_ELIM_THM]
THEN EXISTS_TAC`&0` THEN EXISTS_TAC`&1` THEN REDUCE_VECTOR_TAC THEN REAL_ARITH_TAC;
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[AFF_GE_EQ_AFFINE_HULL;AFFINE_HULL_1;SUBSET;IN_SING;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`&1 %A=A`]]);;

let remark01_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (u:real^3) (w:real^3) (v1:real^3) (w1:real^3) a:real.
FAN(x,V,E)  /\ {v1,w1} IN E /\ {u,w} IN E /\ ~({u,w} = {v1,w1})
/\ &0 <a /\ a< &1 
==> ~((&1 - a) % u + a % w IN {v1, w1:real^3})`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w1:real^3`;`v1:real^3`]
THEN FIND_ASSUM MP_TAC `{v1,w1:real^3} IN E`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN STRIP_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` v1:real^3`;`w1:real^3`]
THEN MRESA_TAC remark012_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` u:real^3`;`w:real^3`;`a:real`]
THEN ASM_TAC THEN SET_TAC[]);;

let remark0_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) e:real^3->bool.
FAN(x,V,E)/\ e IN E 
==> {x} INTER e={}`,
REPEAT STRIP_TAC THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"A") THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool) `;`(E:(real^3->bool)->bool)`;`(e:real^3->bool)`]expand_edge_graph_fan)
THEN RESA_TAC
THEN REMOVE_THEN "A" MP_TAC
THEN RESA_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`v:real^3`]
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC THEN SET_TAC[]);;


let case2_proof_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ {v,w1} IN E 
/\ &0 < azim x u w v /\ azim x u w v < pi
/\ &0<a /\ a< &1
/\ (&1-a)%u+ a%w=ua
/\ ~(ua=w)
/\ ~collinear {x,w,ua}
/\ (aff_gt {x} {v, ua} INTER
      {v | ?e. e IN E /\ v IN aff_ge {x} e} =     {})
==> (!e. e IN (E DELETE {u,w}) ==> aff_ge {x} e INTER aff_ge {x} {ua,w} = aff_ge {x} (e INTER {ua,w}))`,
REPEAT STRIP_TAC
THEN FIND_ASSUM MP_TAC `~collinear {x,w,ua:real^3}`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN DISCH_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`u:real^3`]
THEN FIND_ASSUM MP_TAC `e IN E DELETE {u,w:real^3}`
THEN REWRITE_TAC[DELETE;IN_ELIM_THM;SET_RULE`~(e={u,w:real^3})<=> ~({u,w:real^3}=e)`]
THEN DISCH_TAC
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool) `;`(E:(real^3->bool)->bool)`;`(e:real^3->bool)`]expand_edge_graph_fan)
THEN RESA_TAC
THEN MRESA_TAC lemma_proof_fan [`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (u:real^3)`;`(w:real^3)`;`(v':real^3)`;`(w':real^3)`;`a:real`]
THEN MRESA_TAC remark01_fan [`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (u:real^3)`;`(w:real^3)`;`(v':real^3)`;`(w':real^3)`;`a:real`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM(th)])
THEN RESA_TAC THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(ua IN e)==> e INTER {ua,w:real^3} SUBSET {w}`)
THEN RESA_TAC
THEN MRESA_TAC properties1_inside_fan[`(x:real^3)`;` (u:real^3)`;`(w:real^3)`;]
THEN MRESA_TAC aff_ge1_subset_aff_ge[`x:real^3`;`u:real^3`; `w:real^3`;`ua:real^3`]
THEN MRESA_TAC th3[` (x:real^3)`;`(ua:real^3)`;`(w:real^3)`]
THEN MRESA_TAC remark0_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`e:real^3->bool`]
THEN MRESAL_TAC AFF_GE_MONO_RIGHT [`{x:real^3}`;`e INTER { ua,w:real^3}`;`e:real^3->bool`][DISJOINT;SET_RULE` e INTER {ua,w} SUBSET e`;]
THEN MRESAL_TAC AFF_GE_MONO_RIGHT [`{x:real^3}`;`e INTER { ua,w:real^3}`;`{ua,w:real^3}`][DISJOINT;SET_RULE` e INTER {ua,w} SUBSET {ua,w:real^3}`;]
THEN MRESAL_TAC AFF_GE_MONO_RIGHT [`{x:real^3}`;`{}:real^3->bool`;`{w:real^3}`][DISJOINT;SET_RULE`{x} INTER {w}={} <=> ~(x=w)`;SET_RULE` {} SUBSET {w:real^3}`;]
THEN MATCH_MP_TAC(SET_RULE`C SUBSET A /\ C SUBSET B /\ A INTER B SUBSET C==>A INTER B= C`)
THEN ASM_REWRITE_TAC[]
THEN FIND_ASSUM MP_TAC `FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN;fan7] 
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MP_TAC(ISPECL[`{(u:real^3),w}`;`(e:real^3->bool)`]th))
		   THEN ASM_REWRITE_TAC[UNION; IN_ELIM_THM;]
THEN DISJ_CASES_TAC(SET_RULE`{u,w:real^3} INTER e= {u,w} \/  ~({u,w} INTER e= {u,w})`)
THENL[MP_TAC(SET_RULE`{u,w} INTER e= {u,w:real^3}==> {u,w} SUBSET e`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool) `;`(E:(real^3->bool)->bool)`;`(e:real^3->bool)`]expand_edge_graph_fan)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE` ~(u=w:real^3) ==> ({u,w} = {v',w'}<=> {u,w} SUBSET {v',w'})`)
THEN RESA_TAC
THEN ASM_TAC THEN SET_TAC[];

MP_TAC(SET_RULE` e INTER {ua,w:real^3} SUBSET {w}==>  e INTER { ua,w:real^3} = {w} \/ e INTER { ua,w:real^3}= {}`)
THEN RESA_TAC

THENL[
MP_TAC(SET_RULE`e INTER {ua, w} = {w:real^3}==> w IN e`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~({u, w} INTER e = {u, w:real^3}) ==> {u, w} INTER e PSUBSET {u, w} `)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`{u, w} INTER e PSUBSET {u, w:real^3} /\ w IN e /\ ~(u=w) ==>   {u, w} INTER e SUBSET {w} `)
THEN RESA_TAC
THEN MRESAL_TAC AFF_GE_MONO_RIGHT [`{x:real^3}`;`{u, w} INTER (e:real^3->bool)`;`{w:real^3}`][DISJOINT;SET_RULE`{x} INTER {w}={} <=> ~(x=w)`;SET_RULE` {} SUBSET {w:real^3}`;]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC  THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC  THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC  THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC  THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN POP_ASSUM MP_TAC
 THEN SET_TAC[];

MP_TAC(SET_RULE`e INTER {ua, w:real^3} = {}==> ~(w IN e)`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~({u, w} INTER e = {u, w:real^3}) ==> {u, w} INTER e PSUBSET {u, w} `)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`{u, w} INTER e PSUBSET {u, w:real^3} /\ ~(w IN e) ==>   {u, w} INTER e SUBSET {u} `)
THEN RESA_TAC
THEN MRESAL_TAC AFF_GE_MONO_RIGHT [`{x:real^3}`;`{u, w} INTER (e:real^3->bool)`;`{u:real^3}`][DISJOINT;SET_RULE`{x} INTER {u}={} <=> ~(x=u)`;SET_RULE` {} SUBSET {w:real^3}`;]
THEN STRIP_TAC
THEN SUBGOAL_THEN `aff_ge {x} e INTER aff_ge {x} {ua, w} SUBSET aff_ge {x} {u} INTER aff_ge {x} {ua, w:real^3}`ASSUME_TAC
THENL[
POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC  THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC  THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC  THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC  THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC  
THEN REMOVE_ASSUM_TAC THEN POP_ASSUM MP_TAC THEN SET_TAC[];
MRESA_TAC properties_inside_collinear1_fan[`x:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[]]]]);;






let inequality1_not0_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) (w:real^3) d:real a:real.
FAN(x,V,E) /\ {v,u} IN E/\ {u,w} IN E /\ ~(coplanar{x,v,u,w}) /\ &0 < d
/\ &0<a /\ a < &1 
==>
?(h:real). a <h /\  h<= &1
/\ (!t:real. a <= t /\  t<h
==> (!(s:real). &0 <= s /\ s <= &1 
==> s * inv(norm((&1-s)%v+s%((&1-t)%u+ t%w)-x))* norm( ((&1-a)%u + a %w)-((&1-t)%u+ t%w))< d  ))`,
REPEAT STRIP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`((&1-a)%u + a %w)-((&1-t)%u+ t%w)=(t-a)%(u-w):real^3`;NORM_MUL]
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (v:real^3)`;` (u:real^3)`;` (w:real^3)`]separate_point_convex_fan)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"A")
THEN EXISTS_TAC`min (a+(h:real)*(d:real) *inv(norm(u-w:real^3))) (&1)`
THEN MP_TAC(ISPECL[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] remark1_fan)
THEN RESA_TAC 
THEN  MP_TAC(ISPECL[`u:real^3`;`w:real^3`]imp_norm_gl_zero_fan)
THEN RESA_TAC 
THEN MP_TAC(ISPECL[`(h:real)*(d:real)`;`inv(norm(u-w:real^3))`] REAL_LT_MUL)
THEN RESA_TAC
THEN STRIP_TAC
THENL[
MP_TAC(ISPECL[`h:real`;`d:real`] REAL_LT_MUL)
THEN RESA_TAC
THEN ASM_TAC
THEN REAL_ARITH_TAC;
STRIP_TAC
THENL[
REAL_ARITH_TAC;
REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0 <(h:real)==> ~(h= &0)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(h:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`(t:real)< min (a+(h:real) * (d:real) * inv (norm (u - w:real^3))) (&1)/\ min (a+h * d * inv (norm (u - w))) (&1)<= &1 /\ &0 <a /\ a <= t ==> &0 <= t /\ t <= &1 /\ &0 <= t-a /\ t - a<= &1/\ (t:real)-a < (h * d * inv (norm (u - w:real^3)))`)
THEN ASM_REWRITE_TAC[REAL_ARITH`min (a+h * d * inv (norm (u - w))) (&1)<= &1`]
THEN STRIP_TAC
THEN MP_TAC(ISPECL[`(v:real^3)`;` (u:real^3) `;`(w:real^3) `;`(t:real) `;`s:real`]expansion_convex_fan)
THEN RESA_TAC
THEN REMOVE_THEN "A"(fun th -> MP_TAC(ISPEC`(&1-s)%v+s%((&1-t)%u+ t%w):real^3`th))
THEN RESA_TAC
THEN MP_TAC(ISPECL[`h:real`;`norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3)`]REAL_LT_INV2)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0 <h /\ (h:real)< norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3)==> &0 <= norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3)`REAL_LE_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`t-a:real`REAL_ABS_REFL)
THEN RESP_TAC
THEN MP_TAC(ISPECL[`inv(norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3))`;
`inv(h:real)`;`t:real-a`;`(h * d * inv (norm (u - w:real^3)))`;]REAL_LT_MUL2)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`u:real^3`;`w:real^3`]imp_norm_not_zero_fan)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(norm(u-w:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`u-w:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(&0 =norm(u-w:real^3)) /\ &0 <= norm(u-w:real^3)==> &0 <norm(u-w:real^3)`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`inv (norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3)) * (t:real-a)`;`inv (h:real) * h * d * inv (norm (u - w:real^3))`;`norm (u - w:real^3)`]REAL_LT_RMUL)
THEN ASM_REWRITE_TAC[REAL_ARITH`(A*B*C*D)*E=(A*B)*C*(D*E)`]
THEN REDUCE_ARITH_TAC
THEN STRIP_TAC
THEN MP_TAC(ISPECL[`inv (norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3))`;`(t:real-a)`;] REAL_LE_MUL)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`inv (norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3))*(t-a:real)`;`norm(u-w:real^3)`] REAL_LE_MUL)
THEN RESA_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(s= &1) \/ ~((s:real)= &1)`)
THENL[
POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN REDUCE_ARITH_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`A*B*C=(A*B)*C:real`;VECTOR_ARITH`(&1 - s) % v + s % ((&1 - t) % u + t % w) - x=((&1 - s) % v + s % ((&1 - t) % u + t % w))- x :real^3`];
MP_TAC(REAL_ARITH`~(s= &1) /\ s<= &1==> (s:real)< &1`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`s:real`;`&1`;`(inv (norm (((&1 - s) % v + s % ((&1 - t) % u + t % w)) - x:real^3))*(t-a:real)) * norm(u-w:real^3)`;
`d:real`]REAL_LT_MUL2)
THEN REDUCE_ARITH_TAC
THEN RESA_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`A*d*B*C=A*(d*B)*C:real`;VECTOR_ARITH`(&1 - s) % v + s % ((&1 - t) % u + t % w) - x=((&1 - s) % v + s % ((&1 - t) % u + t % w))- x :real^3`]]]]);;


let bounded_convex1_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) w:real^3.
FAN(x,V,E) /\ {v,u} IN E /\ {u,w} IN E
==> ?(h:real). &0< h /\ (!(y:real^3).  y IN convex hull{v,u,w} ==> norm(y-x)<h )`,
REPEAT STRIP_TAC
THEN SUBGOAL_THEN `FINITE {(v:real^3),(u:real^3),w}` ASSUME_TAC
THENL(*1*)[ SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY;
                 IN_INSERT; NOT_IN_EMPTY];
MP_TAC(ISPEC`{(v:real^3),(u:real^3),w}`FINITE_IMP_COMPACT_CONVEX_HULL)
THEN POP_ASSUM (fun th-> REWRITE_TAC[th;])
THEN DISCH_TAC
THEN ASSUME_TAC(ISPEC`x:real^3` norm_origin_fan )
THEN ASSUME_TAC(SET_RULE`convex hull {(v:real^3), u,w} SUBSET (:real^3)`)
THEN MP_TAC(ISPECL[`(\(y:real^3). lift(norm(y-x:real^3)))`;`(:real^3)`;`convex hull {(v:real^3),(u:real^3),w}`]CONTINUOUS_ON_SUBSET)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(\(y:real^3). lift(norm(y-x:real^3)))`;`convex hull {(v:real^3),(u:real^3),w}`]COMPACT_CONTINUOUS_IMAGE)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED;bounded;IMAGE;IN_ELIM_THM]
THEN ONCE_REWRITE_TAC[TAUT`A/\B<=>B/\A`;]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"A")
THEN SUBGOAL_THEN `(?x':real^3. x' IN convex hull {v, u,w} /\
        lift (norm (v - x:real^3)) = lift (norm (x' - x)))`ASSUME_TAC
THENL[
EXISTS_TAC`v:real^3`
THEN SIMP_TAC[CONVEX_HULL_3; IN_ELIM_THM;]
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&0`
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC;
SUBGOAL_THEN `&0< (a:real)` ASSUME_TAC
THENL[
REMOVE_THEN "A" (fun th-> MP_TAC(ISPEC`lift (norm (v - (x:real^3)))`th))
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[NORM_LIFT;REAL_ABS_NORM ]
THEN MP_TAC(ISPECL[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`] remark1_fan)
THEN RESA_TAC 
THEN MP_TAC(ISPECL[`v:real^3`;`x:real^3`]imp_norm_not_zero_fan)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`v-x:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(&0 =norm(v-x:real^3)) /\ &0 <= norm(v-x:real^3)==> &0 <norm(v-x:real^3)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
EXISTS_TAC`&2 * a:real`
THEN STRIP_TAC
THENL[
REPEAT STRIP_TAC
THEN SUBGOAL_THEN `(?x':real^3. x' IN convex hull {v, u,w} /\
        lift (norm (y - x:real^3)) = lift (norm (x' - x)))`ASSUME_TAC
THENL[
EXISTS_TAC`y:real^3`
THEN ASM_SIMP_TAC[];
REMOVE_THEN "A"(fun th-> MP_TAC(ISPEC`lift (norm (y - x:real^3))`th)) 
THEN ASM_REWRITE_TAC[NORM_LIFT;REAL_ABS_NORM ]
THEN ASM_TAC
THEN REAL_ARITH_TAC];

ASM_TAC
THEN REAL_ARITH_TAC]]]]);;

let inequaility2_not0_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) (w:real^3) d:real a:real.
FAN(x,V,E) /\ {v,u} IN E/\ {u,w} IN E /\ ~(coplanar{x,v,u,w}) /\ &0 < d
/\ &0 <a /\ a< &1
==>
?(h:real). a <h /\  h<= &1
/\ (!t:real. a <= t /\  t<h
==> (!(s:real). &0 <= s /\ s <= &1 
==> (norm(inv(norm((&1-s)%v+s%((&1-a)%u+ a%w)-x))%((&1-s)%v+s%((&1-a)%u+ a%w)-x) - inv(norm((&1-s)%v+s%((&1-t)%u+ t%w)-x))% ((&1-s)%v+s%((&1-a)%u+ a%w)-x)))< d  ))`,

REWRITE_TAC[VECTOR_ARITH`A%B-C%B=(A-C)%B`;NORM_MUL]
THEN REPEAT STRIP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`u-((&1-t)%u+ t%w)=t%(u-w):real^3`;NORM_MUL]
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (v:real^3)`;` (u:real^3)`;` (w:real^3)`]separate_point_convex_fan)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"A")
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (v:real^3) `;`(u:real^3) `;`(w:real^3)`]origin_point_not_in_convex_fan)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"B")
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;`(u:real^3)`;`w:real^3`]bounded_convex1_fan)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"C")
THEN EXISTS_TAC`min (a+h*h*(inv (h'))* inv(norm(u-w:real^3))*d:real) (&1)`
THEN MP_TAC(ISPECL[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] remark1_fan)
THEN RESA_TAC 
THEN MP_TAC(ISPECL[`u:real^3`;`w:real^3`]imp_norm_not_zero_fan)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`u-w:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(&0 =norm(u-w:real^3)) /\ &0 <= norm(u-w:real^3)==> &0 <norm(u-w:real^3)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(norm(u-w:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0 <(h:real)==> ~(h= &0)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(h:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0 <(h':real)==> ~(h'= &0)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(h':real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`norm (u-w:real^3)`REAL_LE_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`norm (u-w:real^3)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(h:real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0 <inv (h:real)==> &0 <= inv(h)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(h':real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0 <inv (h':real)==> &0 <= inv(h')`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0 <a /\ a< &1==> &0 <= a/\ a <= &1`)
THEN RESA_TAC
THEN STRIP_TAC
THENL(*1*)[
MP_TAC(ISPECL[`(h:real)`;`inv (h':real) `]REAL_LT_MUL)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(h:real)`;`(h:real)*inv (h':real) `]REAL_LT_MUL)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(h:real)*(h:real)*inv (h':real)`;`inv(norm(u-w:real^3))`]REAL_LT_MUL)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(h:real)*(h:real)*inv (h':real) *inv(norm(u-w:real^3))`;`d:real`]REAL_LT_MUL)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`a:real < &1 `
THEN REAL_ARITH_TAC;(*1*)

STRIP_TAC
THENL(*2*)[
REAL_ARITH_TAC;(*2*)
REPEAT STRIP_TAC
THEN MP_TAC(ISPECL[`(v:real^3)`;` (u:real^3)`;`w:real^3`;`a:real`;` s:real`]expansion_convex_fan)
THEN RESA_TAC
THEN REMOVE_THEN "C"(fun th->MP_TAC(ISPEC`(&1 - s) % v + s % ((&1 - a) % u + a % w):real^3`th))
THEN REWRITE_TAC[VECTOR_ARITH`(A+B)-C=A+B-C:real^3`]
THEN RESA_TAC
THEN SUBGOAL_THEN`~(x=(&1 - s) % v + s % ((&1 - a) % u + a % w):real^3)` ASSUME_TAC
THENL(*3*)[
ASM_TAC THEN SET_TAC[];(*3*)
MP_TAC(ISPECL[`(&1 - s) % v + s % ((&1 - a) % u + a % w):real^3`;`x:real^3`]imp_norm_not_zero_fan)
THEN REWRITE_TAC[VECTOR_ARITH`(A+B)-C=A+B-C:real^3`]
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`(&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(norm((&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3)= &0) /\ &0 <= norm((&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3)==> &0 <norm((&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(norm((&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(norm((&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3))`REAL_LE_INV)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`(t:real)< min (a+h * h * inv h' * inv (norm (u - w)) * d) (&1)/\ min (a+h * h * inv h' * inv (norm (u - w)) * d) (&1)<= &1 /\ &0<a /\ a<=t ==> &0<= t-a /\ &0<= t /\ t<= &1/\ (t:real)-a< (h * h * inv h' * inv (norm (u - w:real^3)) * d) `)
THEN ASM_REWRITE_TAC[REAL_ARITH`min (a+h * h * inv h' * inv (norm (u - w:real^3)) * d) (&1)<= &1`]
THEN STRIP_TAC
THEN MP_TAC(ISPECL[`norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x:real^3) `;
`(h':real)`;`t-a:real`;`(h * h * inv h' * inv (norm (u - w:real^3)) * d)`;]REAL_LT_MUL2)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= &0/\ &0<= &1`]
THEN USE_THEN "A" (fun th-> MP_TAC(ISPEC`((&1 - s) % v + s % ((&1 - a) % u + a % w):real^3)`th))
THEN REWRITE_TAC[VECTOR_ARITH`(A+B)-C=A+B-C:real^3`]
THEN ASM_REWRITE_TAC[]
THEN DISCH_TAC
THEN MP_TAC(ISPECL[`h:real`;`norm ((&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3)`]REAL_LT_INV2)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`norm((&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3)`;`t-a:real`]REAL_LE_MUL)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`inv(norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x:real^3))`;`inv(h:real)`;`norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x:real^3)*(t:real-a)`;`(h':real)*(h * h * inv h' * inv (norm (u - w:real^3)) * d)`;]REAL_LT_MUL2)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(v:real^3)`;` (u:real^3) `;`(w:real^3) `;`(t:real) `;`s:real`]expansion_convex_fan)
THEN ASM_REWRITE_TAC[]
THEN DISCH_TAC
THEN USE_THEN "A" (fun th-> MP_TAC(ISPEC`(&1 - s) % v + s % ((&1 - t) % u + t % w):real^3`th))
THEN POP_ASSUM(fun th-> REWRITE_TAC[th; VECTOR_ARITH`(A+B)-C=A+B-C:real^3`])
THEN DISCH_TAC
THEN MP_TAC(ISPECL[`h:real`;`norm ((&1 - s) % v + s % ((&1 - t) % u + t % w)-x:real^3)`]REAL_LT_INV2)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`inv(norm((&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3))`;`norm((&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3)*(t-a:real)`]REAL_LE_MUL)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`((&1 - s) % v + s % ((&1 - t) % u + t % w)-x:real^3)`NORM_POS_LE)
THEN MP_TAC(ISPEC`norm ((&1 - s) % v + s % ((&1 - t) % u + t % w)-x:real^3)`REAL_LE_INV)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`inv(norm ((&1 - s) % v + s % ((&1 - t) % u + t % w)-x:real^3))`;`inv(h:real)`;`inv(norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x:real^3)) * norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x:real^3)*(t-a:real)`;`inv(h:real)*(h':real)*(h * h * inv h' * inv (norm (u - w:real^3)) * d)`;]REAL_LT_MUL2)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`norm (u - w:real^3)`;
`inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x:real^3)) *
      inv (norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x)) *
      norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x) *
      (t-a)`;`inv h * inv h * h' * h * h * inv h' * inv (norm (u - w:real^3)) * d`;]REAL_LT_LMUL)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[REAL_ARITH`A*B*C*D*E=(B*C)*E*A*D`]
THEN ONCE_REWRITE_TAC[REAL_ARITH`A*B*C*D*E=(A*B)*E*(C*D)`]
THEN ONCE_REWRITE_TAC[REAL_ARITH`A*B*C*D*E=(A*B)*E*(C*D)`]
THEN ONCE_REWRITE_TAC[REAL_ARITH`A*B*C*D=(A*B)*C*D`]
THEN ONCE_REWRITE_TAC[REAL_ARITH`A*B*C*D*E=(A*B)*(C*D)*E`]
THEN ONCE_REWRITE_TAC[REAL_ARITH`(A*B*C)*D*E=A*B*(C*D)*E`]
THEN ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B*C)*D=A*B*(C*D)`]
THEN REDUCE_ARITH_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`((A*B)*C*D)*E=(A*C)*(B*D)*E`]
THEN REDUCE_ARITH_TAC
THEN REWRITE_TAC[REAL_ARITH`((A*B)*C)*D*E=(A*B)*(C*D)*E`]
THEN ASSUME_TAC(ISPECL[`u:real^3`;`w:real^3`]NORM_SUB)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN MP_TAC(ISPECL[`t-a:real`;`w-u:real^3`]NORM_MUL)
THEN MP_TAC(ISPEC`t-a:real`REAL_ABS_REFL)
THEN RESP_TAC
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN ASM_REWRITE_TAC[REAL_ARITH`(A*B)*C*D=C*A*(B*D)`]
THEN REDUCE_ARITH_TAC
THEN DISCH_THEN(LABEL_TAC"BA")
THEN MP_TAC(ISPECL[`(v:real^3)`;` (u:real^3) `;`(w:real^3) `;`(t:real) `;`s:real`]expansion_convex_fan)
THEN RESA_TAC
THEN SUBGOAL_THEN`~(x=(&1 - s) % v + s % ((&1 - t) % u + t % w):real^3)` ASSUME_TAC
THENL(*4*)[
POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"MA")
THEN STRIP_TAC
THEN REMOVE_THEN"MA" MP_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN ASM_REWRITE_TAC[];(*4*)
MP_TAC(ISPECL[`(&1 - s) % v + s % ((&1 - t) % u + t % w):real^3`;`x:real^3`]imp_norm_not_zero_fan)
THEN REWRITE_TAC[VECTOR_ARITH`(A+B)-C=A+B-C:real^3`]
THEN RESA_TAC
THEN MP_TAC(ISPECL[`norm((&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3)`;`norm((&1 - s) % v + s % ((&1 - t) % u + t % w)-x:real^3)`]REAL_SUB_INV)
THEN RESP_TAC
THEN ASM_REWRITE_TAC[REAL_ABS_ABS;REAL_ABS_DIV;REAL_ABS_MUL;REAL_ABS_NORM;real_div;REAL_INV_MUL;
REAL_ARITH`(A*B*C)*D=A*C*(B*D)`]
THEN REDUCE_ARITH_TAC
THEN MP_TAC(ISPECL[`((&1 - s) % v + s % ((&1 - t) % u + t % w)-x:real^3)`;`((&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3)`]REAL_ABS_SUB_NORM)
THEN REWRITE_TAC[VECTOR_ARITH`
((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)-((&1 - s) % v + s % ((&1 - a) % u + a % w) - x)=s%(t-a)%(w-u) :real^3`]
THEN ONCE_REWRITE_TAC[NORM_MUL]
THEN MP_TAC(ISPEC`s:real`REAL_ABS_REFL)
THEN RESP_TAC
THEN DISCH_TAC
THEN MP_TAC(ISPECL[`abs
 (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x:real^3) -
  norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x))`;`s * norm ((t-a) % (w - u):real^3)`;`inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x:real^3))`]REAL_LE_RMUL)
THEN RESA_TAC
THEN REMOVE_THEN "BA" MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(s= &1)\/ ~((s:real) = &1)`)
THENL(*5*)[
ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC;(*5*)

MP_TAC(REAL_ARITH`~(s= &1) /\ s<= &1==> (s:real)< &1`)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`((t-a) % (w - u):real^3)`NORM_POS_LE)
THEN MP_TAC(ISPECL[`norm ((t-a) % (w - u):real^3)`;`inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x:real^3))`]REAL_LE_MUL)
THEN RESA_TAC
THEN REPEAT STRIP_TAC
THEN MP_TAC(ISPECL[`s:real`;`&1`;`norm ((t-a) % (w - u)) *
     inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x:real^3))`;
`d:real`]REAL_LT_MUL2)
THEN REDUCE_ARITH_TAC
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC]]]]]);;

let exists_point_small_edges_not0_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) (w:real^3) d:real a:real.
FAN(x,V,E) /\ {v,u} IN E/\ {u,w} IN E /\ ~(coplanar{x,v,u,w}) /\ &0 < d
/\ &0 <a /\ a< &1
==>
?(h:real). a <h /\  h<= &1
/\ (!t:real. a <= t /\  t<h
==> (!(s:real). &0 <= s /\ s <= &1 
==> norm(inv(norm((&1-s)%v+s%((&1-a)%u+ a%w)-x))%((&1-s)%v+s%((&1-a)%u+ a%w)-x) - inv(norm((&1-s)%v+s%((&1-t)%u+ t%w)-x))% ((&1-s)%v+s%((&1-t)%u+ t%w)-x))< d  ))`,

REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0< (d:real)==> &0< d/ &2`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (v:real^3) `;`(u:real^3) `;`(w:real^3)`]origin_point_not_in_convex_fan)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (v:real^3) `;`(u:real^3) `;`(w:real^3)`;` (d:real)/ &2`;`a:real`]inequaility2_not0_fan)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"A")
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;`(E:(real^3->bool)->bool) `;`(v:real^3)`;` (u:real^3)`;` (w:real^3)`;` (d:real)/ &2`;`a:real`]inequality1_not0_fan)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"B")
THEN EXISTS_TAC`min (h:real) (h':real)`
THEN STRIP_TAC
THENL[ASM_TAC
THEN REAL_ARITH_TAC;
STRIP_TAC
THENL[
ASM_TAC
THEN REAL_ARITH_TAC;
REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`t<  min (h:real) (h':real)==> t< (h:real) /\ t< (h':real)`)
THEN RESA_TAC
THEN REMOVE_THEN"B" (fun th-> MP_TAC(ISPEC`t:real`th))
THEN RESA_TAC
THEN POP_ASSUM  (fun th-> MP_TAC(ISPEC`s:real`th))
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"B")
THEN REMOVE_THEN"A" (fun th-> MP_TAC(ISPEC`t:real`th))
THEN RESA_TAC
THEN POP_ASSUM  (fun th-> MP_TAC(ISPEC`s:real`th))
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"A")
THEN ASSUME_TAC(ISPEC`((&1 - s) % v + s % ((&1 - t) % u + t % w)-x:real^3)`NORM_POS_LE)
THEN MP_TAC(ISPEC`norm ((&1 - s) % v + s % ((&1 - t) % u + t % w)-x:real^3)`REAL_LE_INV)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`s *
      inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x:real^3)):real`;`(((&1 - a) % u + a % w) - ((&1 - t) % u + t % w)):real^3`]NORM_MUL)
THEN REWRITE_TAC[REAL_ABS_MUL]
THEN MP_TAC(ISPEC`s:real`REAL_ABS_REFL)
THEN RESP_TAC
THEN MP_TAC(ISPEC`inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x:real^3)):real`REAL_ABS_REFL)
THEN RESP_TAC
THEN REWRITE_TAC[REAL_ARITH`(A*B)*C=A*B*C`]
THEN DISCH_TAC
THEN REMOVE_THEN"B"MP_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN REWRITE_TAC[VECTOR_ARITH`(s * inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x))) %
  (((&1 - a) % u + a % w) - ((&1 - t) % u + t % w))= inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) % ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x)- inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
      ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x):real^3`]

THEN DISCH_TAC
THEN MP_TAC(ISPECL[`norm(inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) % ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x)- inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
      ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x):real^3)`;`(d:real)/ &2`;
`norm
      (inv (norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x)) % ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x) -
       inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
       ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x):real^3)`;`(d:real)/ &2`;
]REAL_LT_ADD2)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`d / &2 + d/ &2 = d`]
THEN MP_TAC(ISPECL[`(inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) % ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x)- inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
      ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x):real^3)`;
`(inv (norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x)) % ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x) -
       inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
       ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x):real^3)`]NORM_TRIANGLE)
THEN REWRITE_TAC[VECTOR_ARITH`(inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) % ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x)- inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
      ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x):real^3)+
(inv (norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x)) % ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x) -
       inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
       ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x):real^3)=(inv (norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x)) % ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x) -
  inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
  ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) `]
THEN REAL_ARITH_TAC]]);;


(*********************************)


let separate1_sphere_not0_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) (w:real^3) (v1:real^3) (u1:real^3) a:real.

FAN(x,V,E) /\  {v,u} INTER {v1,u1}={} /\ {v,u} IN E/\ {u,w} IN E /\ {v1,u1} IN E
/\ ~(coplanar{x,v,u,w})
/\ (&0< azim x u w v ) /\ (azim x u w v <pi)
/\ &0 < a /\  a < &1 /\
         aff_gt {x} {v, (&1 - a) % u + a % w} INTER
         {v | ?e. e IN E /\ v IN aff_ge {x} e} =
         {}
==>
?h:real.
(a < h) /\ (h<= &1)
/\ 
(!t:real. a<t /\ t<h 
==> aff_gt{x} {v,(&1-t)%u+ t % w} INTER aff_ge {x} {v1, u1} INTER ballnorm_fan x={})`,
REPEAT STRIP_TAC 
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`] 
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] 
THEN MRESA_TAC exist_close1_fan[`(x:real^3)` ;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (v:real^3)`;`(u:real^3)`;`w:real^3`;`(v1:real^3)`;` (u1:real^3)`;`a:real`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THA")
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded[`(x:real^3)` ;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (v:real^3)`;`(u:real^3)`;`(w:real^3)`;]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"A" )
THEN MRESA_TAC exists_point_small_edges_not0_fan [`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(v:real^3)`;`(u:real^3) `;`(w:real^3)`;`h:real`;`a:real`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"B" )
THEN EXISTS_TAC`min (a+((h'-a:real)/ &2)) (&1:real)`
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*1*)[
MP_TAC(REAL_ARITH`a< h':real==> &0< (h'-a)/ &2`)
THEN ASM_REWRITE_TAC[]
THEN FIND_ASSUM MP_TAC`a:real< &1`
THEN REAL_ARITH_TAC;(*1*)
STRIP_TAC
THENL(*2*)[
ASM_TAC
THEN REAL_ARITH_TAC;(*2*)
REPEAT STRIP_TAC
THEN REWRITE_TAC[EXTENSION]
THEN GEN_TAC
THEN EQ_TAC
THENL(*3*)[
STRIP_TAC
THEN REWRITE_TAC[EMPTY;IN_ELIM_THM]
THEN SUBGOAL_THEN `!y1:real^3.
          y1 IN aff_gt {x} {v, (&1-t)%u+(t:real)% (w:real^3)} INTER ballnorm_fan x
==> ?y2:real^3.        y2 IN aff_ge {x:real^3} {(v:real^3), ((&1 - a) % u + a % w:real^3)} INTER ballnorm_fan x /\
          dist (y1,y2)<(h:real)` ASSUME_TAC
THENL(*4*)[
MP_TAC(REAL_ARITH`&0<a /\ a < &1 /\ a <(t:real)/\ a< h' /\ t < min (a+(h'-a:real)/ &2) (&1:real)==> &0<t /\ t< &1 /\ &0 <= t /\ t<= &1 /\ t< h'/\ &0<=a /\ a<= &1/\ a<=t`)
THEN RESA_TAC
THEN USE_THEN"A" (fun th-> MP_TAC(ISPEC`a:real`th))
THEN REMOVE_THEN"A" (fun th-> MP_TAC(ISPEC`t:real`th))
THEN RESA_TAC
THEN RESA_TAC
THEN REPEAT STRIP_TAC
THEN MRESA_TAC same_projective_sphere_gt_fan [`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(v:real^3)`;`(u:real^3) `;`(w:real^3)`;` (t:real) `;`(y1:real^3)`]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - a) % (u:real^3) + a % (w:real^3)`;]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - t) % (u:real^3) + t % (w:real^3)`;]
THEN MRESA_TAC AFF_GE_1_2[`(x:real^3)` ;` (v:real^3)`;`((&1 - a) % u + a % w:real^3) `;]
THEN ASM_REWRITE_TAC[INTER;IN_ELIM_THM;dist]
THEN MRESA_TAC expansion_convex_fan[`(v:real^3)`;` (u:real^3)`;`w:real^3`;`a:real`;` s:real`;]
THEN MRESA_TAC origin_point_not_in_convex_fan[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;`(u:real^3)`;`w:real^3`]
THEN SUBGOAL_THEN`~(x=(&1-s)%v+s%((&1 - a) % u + a % w):real^3)` ASSUME_TAC
THENL(*5*)[
POP_ASSUM MP_TAC
THEN MATCH_MP_TAC MONO_NOT
THEN RESA_TAC;(*5*)
MP_TAC(ISPECL[`x:real^3`;`(&1 - s) % v + s % ((&1 - a) % u + a % w):real^3`]imp_norm_not_zero_fan)
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[NORM_SUB]
THEN REWRITE_TAC[VECTOR_ARITH`(A+B)-C=A+B-C:real^3`]
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`(&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(&0 =norm((&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3)) /\ &0 <= norm((&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3)==> &0 <norm((&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(norm((&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(norm((&1 - s) % v + s % ((&1 - a) % u + a % w)-x:real^3))`REAL_LE_INV)
THEN RESA_TAC
THEN EXISTS_TAC`inv (norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x)) %
      ((&1 - s) % v + s %  ((&1 - a) % u + a % w) - x) +x:real^3`
THEN STRIP_TAC
THENL(*6*)[
STRIP_TAC
THENL(*7*)[
EXISTS_TAC`&1 - inv (norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x:real^3)) `
THEN EXISTS_TAC`inv (norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x:real^3)) * (&1 - s)`
THEN EXISTS_TAC`inv (norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x:real^3)) * (s)`
THEN REWRITE_TAC[VECTOR_ARITH`A%(B%C + a%b-c)+c=(&1-A)%c+(A*B)%C+(A*a)%b:real^3`]
THEN ASM_REWRITE_TAC[REAL_ARITH`&1-A+A*(&1-C)+A*C= &1`]
THEN STRIP_TAC
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REAL_ARITH_TAC;(*7*)
REWRITE_TAC[ballnorm_fan;IN_ELIM_THM;dist;VECTOR_ARITH`A-(B+A)=(--B):real^3`;NORM_NEG;NORM_MUL]
THEN MP_TAC(ISPEC`inv (norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x:real^3))`REAL_ABS_REFL)
THEN RESP_TAC
THEN ASM_REWRITE_TAC[]](*7*);(*6*)

ASM_REWRITE_TAC[VECTOR_ARITH`inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
  ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x) +
  x -
  (inv (norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x)) % ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x) + x)=
inv (norm ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x)) %
  ((&1 - s) % v + s % ((&1 - t) % u + t % w) - x) -
  (inv (norm ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x)) % ((&1 - s) % v + s % ((&1 - a) % u + a % w) - x) ):real^3`]
THEN REMOVE_THEN "B" (fun th-> MP_TAC(ISPEC`t:real`th))
THEN RESA_TAC
THEN POP_ASSUM (fun th-> MP_TAC(ISPEC`s:real`th))
THEN RESA_TAC
THEN ASM_REWRITE_TAC[NORM_SUB]](*6*)](*5*);(*4*)
POP_ASSUM (fun th->MP_TAC(ISPEC`x':real^3`th))
THEN MP_TAC(SET_RULE` x' IN aff_gt {x} {v, (&1 - t) % u + t % w} INTER aff_ge {x} {v1, u1} INTER
      ballnorm_fan x ==>  x' IN aff_ge {x} {v1, u1:real^3} INTER ballnorm_fan x
/\ x' IN aff_gt {x} {v, (&1 - t) % u + t % w:real^3} INTER ballnorm_fan x`)
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN SIMP_TAC[]
THEN REPEAT STRIP_TAC
THEN REMOVE_THEN "THA" (fun th->  MP_TAC(ISPECL[`y2:real^3`;`x':real^3`]th))
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN SIMP_TAC[dist;NORM_SUB]
THEN REAL_ARITH_TAC](*4*);(*3*)

SET_TAC[]]]]);;


let fan_run_in_small11_not0_is_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 v1:real^3 u1:real^3 a:real.
FAN(x,V,E) /\  {v,u} INTER {v1,u1}={} /\ {v,u} IN E/\ {u,w} IN E /\ {v1,u1} IN E
/\ ~(coplanar{x,v,u,w})
/\ (&0< azim x u w v ) /\ (azim x u w v <pi)
/\ &0 < a /\  a < &1 /\
         aff_gt {x} {v, (&1 - a) % u + a % w} INTER
         {v | ?e. e IN E /\ v IN aff_ge {x} e} =
         {}
==> 
?t1:real. a < t1 /\ t1 <= &1
/\ (!t:real. a< t /\ t< t1 ==> aff_gt{x} {v,(&1-t)%u+ t % w} INTER aff_ge{x} {v1,u1}={})`,

REPEAT STRIP_TAC 
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u1:real^3)`;
` (v1:real^3)`]  
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded[`(x:real^3)` ;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (v:real^3)`;`(u:real^3)`;`(w:real^3)`;]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"BA")
THEN MRESA_TAC separate1_sphere_not0_fan[`(x:real^3)`;` (V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (v:real^3) `;`(u:real^3)`;` (w:real^3)`;` (v1:real^3)`;`(u1:real^3)`;`a:real`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"A")
THEN REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH` a < &1 /\ a < h ==> a < min (h:real) (&1:real)`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`  h <= &1==>  min (h:real) (&1:real)<= &1`)
THEN RESA_TAC
THEN EXISTS_TAC`min (h:real) (&1:real)`
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`  &0< a /\ a < t /\ t<  min (h:real) (&1:real)==> t< h /\ t<= &1 /\ &0 <= t/\ &0<t /\ t< &1`)
THEN RESA_TAC
THEN REMOVE_THEN "A"(fun th-> MRESA1_TAC th `(t:real)`)
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"A")
THEN REMOVE_THEN "BA"(fun th-> MRESA1_TAC th `(t:real)`)
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - t) % (u:real^3) + t % (w:real^3)`;]
THEN MRESA_TAC origin_is_not_aff_gt_fan[`x:real^3`;`v:real^3`;`(&1-t)%u+t%w:real^3`]
THEN REWRITE_TAC[EXTENSION;INTER;IN_ELIM_THM]
THEN GEN_TAC
THEN EQ_TAC
THENL[ 
REPEAT STRIP_TAC
THEN SUBGOAL_THEN`~(x=x':real^3)`ASSUME_TAC
THENL[
POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];
MRESA_TAC imp_norm_not_zero_fan[`x':real^3`;`x:real^3`]
THEN ASSUME_TAC(ISPEC`x'-x:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(norm(x'-x:real^3)= &0) /\ &0 <= norm(x'-x:real^3)==> &0 <norm(x'-x:real^3)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LE_INV `norm(x'-x:real^3)`
THEN MRESA1_TAC REAL_LT_INV `norm(x'-x:real^3)`
THEN MRESA1_TAC REAL_MUL_LINV `norm(x'-x:real^3)`
THEN MRESA_TAC scale_aff_ge_fan[`x:real^3`;`v1:real^3`;`u1:real^3`]
THEN POP_ASSUM(fun th -> MRESA_TAC th [`x':real^3`;`inv(norm(x'-x:real^3))`])
THEN MRESA_TAC scale_aff_gt_fan[`x:real^3`;`v:real^3`;`(&1-t)%u+t%w:real^3`]
THEN POP_ASSUM(fun th -> MRESA_TAC th [`x':real^3`;`inv(norm(x'-x:real^3))`])
THEN SUBGOAL_THEN`inv (norm (x' - x)) % (x' - x) + x IN ballnorm_fan x:real^3->bool` ASSUME_TAC
THENL[
MRESA1_TAC REAL_ABS_REFL `inv(norm(x'-x:real^3))`
THEN ASM_REWRITE_TAC[ballnorm_fan;IN_ELIM_THM;dist;VECTOR_ARITH`B-(A+B)= --A:real^3`;NORM_NEG;NORM_MUL];
ASM_TAC
THEN SET_TAC[]]];

SET_TAC[]]);;

let fan_run_in_small1_not0_is_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 v1:real^3 u1:real^3 a:real.
FAN(x,V,E) /\  {v,u} INTER {v1,u1}={} /\ {v,u} IN E/\ {u,w} IN E /\ {v1,u1} IN E
/\ ~(coplanar{x,v,u,w})
/\ (&0< azim x u w v ) /\ (azim x u w v <pi)
/\ &0 < a /\  a < &1 /\
         aff_gt {x} {v, (&1 - a) % u + a % w} INTER
         {v | ?e. e IN E /\ v IN aff_ge {x} e} =
         {}
/\ (!s. &0 < s /\ s < a
          ==> aff_gt {x} {v, (&1 - s) % u + s % w} INTER
              {v | ?e. e IN E /\ v IN aff_ge {x} e} =
              {})
==> 
(?t1:real. a < t1 /\ t1 <= &1
/\ (!t:real. &0< t /\ t< t1 ==> aff_gt{x} {v,(&1-t)%u+ t % w} INTER aff_ge{x} {v1,u1}={}))`,
REPEAT STRIP_TAC
THEN MRESA_TAC fan_run_in_small11_not0_is_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;` w:real^3`;`v1:real^3`;`u1:real^3`;`a:real`]
THEN EXISTS_TAC`t1:real`
THEN ASM_REWRITE_TAC[]
THEN GEN_TAC
THEN MP_TAC(REAL_ARITH`&0<a /\ a< t1==>((&0<t /\ t<t1) <=> (&0<t /\ t<a)\/ t=a \/ (a<t/\ t<t1:real))`)
THEN RESA_TAC
THEN STRIP_TAC
THEN ASM_TAC THEN SET_TAC[]);;
let fan_run_in_small2_not0_is_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 w1:real^3 a:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ {v,w1} IN E 
/\ ~coplanar {x,v,u,w}
/\ (&0< azim x u w v ) /\ (azim x u w v <pi)
/\ &0 <a /\ a< &1 
/\(aff_gt {x} {v, (&1 - a) % u + a % w} INTER
      {v | ?e. e IN E /\ v IN aff_ge {x} e} =     {})
/\ (!s. &0 < s /\ s < a
          ==> aff_gt {x} {v, (&1 - s) % u + s % w} INTER
              {v | ?e. e IN E /\ v IN aff_ge {x} e} =
              {})
==> ?t1. a < t1 /\ t1<= &1
/\ (!t:real. &0< t /\ t< t1==> aff_gt{x} {v,(&1-t)%u+ t % w} INTER aff_ge{x} {v,w1}={})`,


REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"MAI MAI")
THEN DISCH_THEN(LABEL_TAC"YEU EM")
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded[`(x:real^3)` ;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (v:real^3)`;`(u:real^3)`;`(w:real^3)`;]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"A")
THEN DISJ_CASES_TAC(SET_RULE`(?h:real. a<h /\ h<= &1/\ azim x v u w1=azim x v u ((&1-h)%u+h%w:real^3))\/ ~(?h:real. a<h /\ h<= &1 /\ azim x v u ((&1-h)%u+h%w:real^3)=azim x v u w1)`)
THENL(*1*)[
POP_ASSUM MP_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`(h:real)`
THEN STRIP_TAC
THENL(*2*)[
ASM_TAC
THEN REAL_ARITH_TAC;(*2*)
STRIP_TAC
THENL(*3*)[
ASM_TAC
THEN REAL_ARITH_TAC;(*3*)
REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0<t:real==> &0<=t`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`(t:real)< (h:real) /\ h <= &1 ==> t < &1`)
THEN RESA_TAC
THEN REMOVE_THEN"A"(fun th-> MRESA1_TAC th `t:real`)
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - t) % (u:real^3) + t % (w:real^3)`;]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w1:real^3)`;
` (v:real^3)`]  
THEN REWRITE_TAC[INTER;EXTENSION;IN_ELIM_THM;]
THEN GEN_TAC
THEN EQ_TAC
THENL(*4*)[

DISCH_TAC
THEN SUBGOAL_THEN `(&1-t)%u+t%w IN aff_ge {x,v} {w1}:real^3->bool` ASSUME_TAC
THENL(*5*)[
POP_ASSUM MP_TAC
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v:real^3`;`w1:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GE_2_1[`x:real^3`;`v:real^3`;`w1:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`(&1-t)%u +t%w:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2 % v + t3 % ((&1 - t) % u + t % w) =
 t1'' % x + t2' % v + t3' % w1<=> t3 % ((&1 - t) % u + t % w) =
 (t1''-t1') % x + (t2'-t2) % v + t3' % w1:real^3`]
THEN MP_TAC(REAL_ARITH`&0<t3:real==> ~(t3= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`(t3:real)`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t3 % ((&1 - t) % u + t % w) = (t1'-t1) % x + (t2'-t2) % v + t3' % w1:real^3 
==> (inv (t3))%(t3 % ((&1 - t) % u + t % w)) = (inv (t3))%((t1'-t1) % x + (t2'-t2) % v + t3' % w1)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`;VECTOR_ARITH`A%(B%X+C%Y+D%Z)=(A*B)%X+(A*C)%Y+(A*D)%Z`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC`t1' + t2' + t3' = &1:real`
THEN FIND_ASSUM MP_TAC`t1 + t2 + t3 = &1:real`
THEN DISCH_TAC 
THEN POP_ASSUM (fun th -> GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SYM(th)])
THEN REWRITE_TAC[REAL_ARITH`t1'' + t2' + t3' = t1' + t2 + t3<=> (t1''-t1') + (t2'-t2) + t3' = t3:real`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
(t1'-t1) + (t2'-t2) + t3' = t3
==> (inv (t3))*((t1'-t1) + (t2'-t2) + t3') = (inv (t3))*t3`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C+D)=A*B+A*C+A*D`]
THEN STRIP_TAC
THEN EXISTS_TAC`inv t3 * (t1' - t1):real`
THEN EXISTS_TAC`inv t3 * (t2' - t2):real`
THEN EXISTS_TAC`inv t3 * t3':real`
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LE_INV
THEN ASM_TAC
THEN REAL_ARITH_TAC;(*5*)

MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`v:real^3`;`(&1 - t) % u + t % w:real^3`;`w1:real^3`;]
THEN MRESA_TAC AZIM_EQ_0_ALT[`x:real^3`;`v:real^3`;`(&1 - t) % u + t % w:real^3`;`w1:real^3`;]
THEN MRESA_TAC AZIM_EQ_ALT[`x:real^3`;`v:real^3`;`u:real^3`;`w1:real^3`;`(&1 - t) % u + t % w:real^3`;]
THEN MRESA_TAC injective_azim_coplanar[`x:real^3`;` v:real^3`;` u:real^3`;` w:real^3`]
THEN POP_ASSUM (fun th->MRESA_TAC th [`h:real`;`t:real`])
THEN ASM_TAC
THEN REAL_ARITH_TAC](*5*)(*4*);
SET_TAC[]]]];(*1*)
POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[NOT_EXISTS_THM]
THEN DISCH_THEN(LABEL_TAC"B")
THEN EXISTS_TAC`&1:real`
THEN ASM_REWRITE_TAC[REAL_ARITH	`&1<= &1`]
THEN REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0<t:real==> &0<=t`)
THEN RESA_TAC
THEN REMOVE_THEN"A"(fun th-> MRESA1_TAC th `t:real`)
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - t) % (u:real^3) + t % (w:real^3)`;]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w1:real^3)`;
` (v:real^3)`]  
THEN REWRITE_TAC[INTER;EXTENSION;IN_ELIM_THM;]
THEN GEN_TAC
THEN EQ_TAC
THENL(*2*)[
DISCH_TAC
THEN SUBGOAL_THEN `(&1-t)%u+t%w IN aff_ge {x,v} {w1}:real^3->bool` ASSUME_TAC
THENL(*3*)[
POP_ASSUM MP_TAC
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v:real^3`;`w1:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GE_2_1[`x:real^3`;`v:real^3`;`w1:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`(&1-t)%u +t%w:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2 % v + t3 % ((&1 - t) % u + t % w) =
 t1'' % x + t2' % v + t3' % w1<=> t3 % ((&1 - t) % u + t % w) =
 (t1''-t1') % x + (t2'-t2) % v + t3' % w1:real^3`]
THEN MP_TAC(REAL_ARITH`&0<t3:real==> ~(t3= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`(t3:real)`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t3 % ((&1 - t) % u + t % w) = (t1'-t1) % x + (t2'-t2) % v + t3' % w1:real^3 
==> (inv (t3))%(t3 % ((&1 - t) % u + t % w)) = (inv (t3))%((t1'-t1) % x + (t2'-t2) % v + t3' % w1)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`;VECTOR_ARITH`A%(B%X+C%Y+D%Z)=(A*B)%X+(A*C)%Y+(A*D)%Z`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC`t1' + t2' + t3' = &1:real`
THEN FIND_ASSUM MP_TAC`t1 + t2 + t3 = &1:real`
THEN DISCH_TAC 
THEN POP_ASSUM (fun th -> GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SYM(th)])
THEN REWRITE_TAC[REAL_ARITH`t1'' + t2' + t3' = t1' + t2 + t3<=> (t1''-t1') + (t2'-t2) + t3' = t3:real`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
(t1'-t1) + (t2'-t2) + t3' = t3
==> (inv (t3))*((t1'-t1) + (t2'-t2) + t3') = (inv (t3))*t3`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C+D)=A*B+A*C+A*D`]
THEN STRIP_TAC
THEN EXISTS_TAC`inv t3 * (t1' - t1):real`
THEN EXISTS_TAC`inv t3 * (t2' - t2):real`
THEN EXISTS_TAC`inv t3 * t3':real`
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LE_INV
THEN ASM_TAC
THEN REAL_ARITH_TAC;(*3*)
MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`v:real^3`;`(&1 - t) % u + t % w:real^3`;`w1:real^3`;]
THEN MRESA_TAC AZIM_EQ_0_ALT[`x:real^3`;`v:real^3`;`(&1 - t) % u + t % w:real^3`;`w1:real^3`;]
THEN MRESA_TAC AZIM_EQ_ALT[`x:real^3`;`v:real^3`;`u:real^3`;`w1:real^3`;`(&1 - t) % u + t % w:real^3`;]
THEN REMOVE_THEN "B"(fun th -> MRESAL1_TAC th `t:real`[DE_MORGAN_THM; REAL_ARITH`~(A<B)<=> B<A\/ B=A`])
THENL(*4*)[REMOVE_THEN"YEU EM"(fun th-> MRESA1_TAC th `t:real`)
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC `{v,w1:real^3} IN E`
THEN SET_TAC[];
REMOVE_THEN"MAI MAI" MP_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC `{v,w1:real^3} IN E`
THEN SET_TAC[];
ASM_TAC
THEN REAL_ARITH_TAC](*4*)];
SET_TAC[]]]);;








let fan_run_in_small3_not0_is_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 w1:real^3 a:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ {u,w1} IN E 
/\ ~coplanar {x,v,u,w} /\ sigma_fan x V E u w=v
/\ (&0< azim x u w v ) /\ (azim x u w v <pi)
/\ &0 <a /\ a< &1 
/\(aff_gt {x} {v, (&1 - a) % u + a % w} INTER
      {v | ?e. e IN E /\ v IN aff_ge {x} e} =     {})
/\ (!s. &0 < s /\ s < a
          ==> aff_gt {x} {v, (&1 - s) % u + s % w} INTER
              {v | ?e. e IN E /\ v IN aff_ge {x} e} =
              {})
==> ?t1. a < t1 /\ t1 <= &1
/\  (!t:real. &0< t /\ t< t1==> aff_gt{x} {v,(&1-t)%u+ t % w} INTER aff_ge{x} {u,w1}={})`,
REPEAT STRIP_TAC
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded[`(x:real^3)` ;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (v:real^3)`;`(u:real^3)`;`(w:real^3)`;]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"A")
THEN EXISTS_TAC`&1:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`&1<= &1`]
THEN REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0<t:real==> &0<=t /\ ~(t= &0)`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`t< &1 ==>t<= &1`)
THEN RESA_TAC
THEN REMOVE_THEN"A"(fun th-> MRESA1_TAC th `t:real`)
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - t) % (u:real^3) + t % (w:real^3)`;]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w1:real^3)`;
` (u:real^3)`]  
THEN REWRITE_TAC[EXTENSION;INTER;IN_ELIM_THM]
THEN GEN_TAC
THEN EQ_TAC
THENL(*1*)[
MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`(&1-t)%u +t%w:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN SUBGOAL_THEN `x' IN (aff_ge {x,u} {w1}:real^3->bool)`ASSUME_TAC
THENL(*2*)[
POP_ASSUM MP_TAC
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`u:real^3`;`w1:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GE_2_1[`x:real^3`;`u:real^3`;`w1:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`t1':real`
THEN EXISTS_TAC`t2':real`
THEN EXISTS_TAC`t3':real`
THEN ASM_REWRITE_TAC[];
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MRESA_TAC inequality3_aim_in_convex_fan
[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;` w:real^3`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th `t:real`)
THEN POP_ASSUM(fun th-> MRESA_TAC th [`t1:real`;`t2:real`;`t3:real`])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"BE")
THEN DISCH_THEN(LABEL_TAC"YEU")
THEN SUBGOAL_THEN`~collinear {x, u, x':real^3}`ASSUME_TAC
THENL(*3*)[
MRESA_TAC continuous_coplanar_fan [`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `t:real`)
THEN ASM_REWRITE_TAC[collinear_fan;]
THEN POP_ASSUM MP_TAC
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[coplanar;aff; AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`x:real^3`
THEN EXISTS_TAC`v:real^3`
THEN EXISTS_TAC`u:real^3`
THEN SUBGOAL_THEN`(x:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL(*4*)[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&0`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;

SUBGOAL_THEN`(v:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL(*5*)[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC`&0`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;
SUBGOAL_THEN`(u:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL(*6*)[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;
SUBGOAL_THEN`((&1 - t) % u + t % w:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL(*7*)[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(REAL_ARITH`&0<t3:real==> ~(t3= &0)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(t3:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REWRITE_TAC[VECTOR_ARITH`t1' % x + t2 % v + t3 % ((&1 - t) % u + t % w) = u' % x + v' % u
<=>t3 % ((&1 - t) % u + t % w) = (u'-t1') % x -t2 % v+v' % u:real^3`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t3 % ((&1 - t) % u + t % w) = (u'-t1) % x -t2 % v+v' % u
==> (inv (t3))%(t3 % ((&1 - t) % u + t % w) ) = (inv (t3))%((u'-t1) % x -t2 % v+v' % u):real^3`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%(B-C+D)=A%B-A%C+A%D`;VECTOR_ARITH`A%B%C=(A*B)%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`(inv t3 * (u' - t1)):real`
THEN EXISTS_TAC`-- (inv t3 * t2):real`
THEN EXISTS_TAC`(inv t3 * v'):real`
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A+(--b)%C+d=A-b%C+d`;REAL_ARITH`inv t3 * (u' - t1') + --(inv t3 * t2) + inv t3 * v'=inv t3 * (t3+(u'+v') - (t1'+ t2+t3))`;REAL_ARITH`A+ &1- &1=A`];(*7*)

ASM_TAC
THEN SET_TAC[]](*7*)](*6*)](*5*)](*4*);(*3*)
 
MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`]
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`u:real^3`;`x':real^3`;`w1:real^3`;]
THEN MRESA_TAC AZIM_EQ_0_ALT[`x:real^3`;`u:real^3`;`x':real^3`;`w1:real^3`;]
THEN MRESA_TAC AZIM_EQ_ALT[`x:real^3`;`u:real^3`;`w:real^3`;`w1:real^3`;`x':real^3`;]
THEN REMOVE_THEN"BE" MP_TAC
THEN REMOVE_THEN"YEU" MP_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN FIND_ASSUM MP_TAC`(sigma_fan x V E u w = (v:real^3))`
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN DISJ_CASES_TAC(SET_RULE`(set_of_edge u V E = {w:real^3}) \/ ~(set_of_edge u V E = {w})`)
THENL(*4*)[
FIND_ASSUM MP_TAC`w1 IN set_of_edge (u:real^3) V E`
THEN POP_ASSUM (fun th-> REWRITE_TAC[(th);IN_SING])
THEN DISCH_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC (ISPECL[`x:real^3`;`u:real^3`;`w:real^3`]AZIM_REFL)
THEN REAL_ARITH_TAC;(*4*)

DISJ_CASES_TAC(SET_RULE`(w1:real^3)=w \/ ~((w1:real^3)=w)`)
THENL(*5*)[
ASM_REWRITE_TAC[]
THEN MP_TAC (ISPECL[`x:real^3`;`u:real^3`;`w:real^3`]AZIM_REFL)
THEN REAL_ARITH_TAC;
MRESA_TAC SIGMA_FAN[`(x:real^3)`;` (V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(u:real^3)`;`(w:real^3)`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `w1:real^3`)
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC]]]];

SET_TAC[]]);;




let fan_run_in_small_not0_is_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 v1:real^3 w1:real^3 a:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ {v1,w1} IN E 
/\ (&0< azim x u w v ) /\ (azim x u w v <pi)
/\ sigma_fan x V E u w = v
/\ &0 <a /\ a< &1 
/\(aff_gt {x} {v, (&1 - a) % u + a % w} INTER
      {v | ?e. e IN E /\ v IN aff_ge {x} e} =     {})
/\ (!s. &0 < s /\ s < a
          ==> aff_gt {x} {v, (&1 - s) % u + s % w} INTER
              {v | ?e. e IN E /\ v IN aff_ge {x} e} =
              {})

==> 
?t1:real. a<t1 /\ t1<= &1
/\ (!t:real. &0< t /\ t< t1==> aff_gt{x} {v,(&1-t)%u+ t % w} INTER aff_ge{x} {v1,w1}={})`,


(let lem=prove(`!x v v1. aff_ge {x} {v1, v}=aff_ge {x} {v, v1}`,
REPEAT STRIP_TAC
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`{v,u}={u,v}`]
THEN SIMP_TAC[]) in

REPEAT STRIP_TAC
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN DISJ_CASES_TAC(SET_RULE`(v1=v)\/ ~(v1=v:real^3)`)
THENL[
MRESA_TAC fan_run_in_small2_not0_is_fan [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v1:real^3`;`u:real^3`;` w:real^3`;`w1:real^3`;`a:real`];

DISJ_CASES_TAC(SET_RULE`(v1=u)\/ ~(v1=u:real^3)`)
THENL[
MRESA_TAC fan_run_in_small3_not0_is_fan[`x:real^3`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` v:real^3`;` v1:real^3`;` w:real^3`;`w1:real^3`;`a:real`];
 DISJ_CASES_TAC(SET_RULE`(w1=v)\/ ~(w1=v:real^3)`)
THENL[
FIND_ASSUM MP_TAC`{v1,w1} IN (E:(real^3->bool)->bool)`
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`{v,u}={u,v}`]
THEN STRIP_TAC
THEN MRESA_TAC fan_run_in_small2_not0_is_fan [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`w1:real^3`;`u:real^3`;` w:real^3`;`v1:real^3`;`a:real`]
THEN EXISTS_TAC`t1:real`
THEN ASM_REWRITE_TAC[lem];
DISJ_CASES_TAC(SET_RULE`(w1=u)\/ ~(w1=u:real^3)`)
THENL[
FIND_ASSUM MP_TAC`{v1,w1} IN (E:(real^3->bool)->bool)`
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`{v,u}={u,v}`]
THEN STRIP_TAC
THEN MRESA_TAC fan_run_in_small3_not0_is_fan[`x:real^3`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` v:real^3`;` w1:real^3`;` w:real^3`;`v1:real^3`;`a:real`]
THEN EXISTS_TAC`t1:real`
THEN ASM_REWRITE_TAC[lem];
SUBGOAL_THEN`{v,u} INTER {v1,w1:real^3}={}` ASSUME_TAC
THENL[
ASM_TAC
THEN SET_TAC[];
MRESA_TAC fan_run_in_small1_not0_is_fan[`x:real^3`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` v:real^3`;` u:real^3`;` w:real^3`;`v1:real^3`;`w1:real^3`;`a:real`]]]]]]));;

let fan_run1_in_small_not0_is_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) (E':(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ E' SUBSET E
/\ (&0< azim x u w v ) /\ (azim x u w v <pi)
/\ sigma_fan x V E u w = v
/\ &0 <a /\ a< &1 
/\(aff_gt {x} {v, (&1 - a) % u + a % w} INTER
      {v | ?e. e IN E /\ v IN aff_ge {x} e} =     {})
/\ (!s. &0 < s /\ s < a
          ==> aff_gt {x} {v, (&1 - s) % u + s % w} INTER
              {v | ?e. e IN E /\ v IN aff_ge {x} e} =
              {})
==>
?h:real. a<h /\ h<= &1
/\ (!s:real. &0< s /\ s< h==> aff_gt{x} {v,(&1-s)%u+ s % w} INTER {v | ?e. e IN E' /\ v IN aff_ge {x} e}={})`,

REPEAT STRIP_TAC 
THEN MRESA_TAC set_edges_is_finite_fan [`x:real^3`;`V:real^3->bool`;`E:(real^3->bool)->bool`]
THEN MRESA_TAC  FINITE_SUBSET [`(E':(real^3->bool)->bool)`;`(E:(real^3->bool)->bool)`]
THEN ABBREV_TAC`n=CARD (E':(real^3->bool)->bool)`
THEN REPEAT(POP_ASSUM MP_TAC)
THEN SPEC_TAC (`(E':(real^3->bool)->bool)`,`(E':(real^3->bool)->bool)`)
THEN SPEC_TAC (`n:num`,`n:num`)
THEN INDUCT_TAC
THENL(*1*)[
REPEAT STRIP_TAC 
THEN MP_TAC(ISPECL[`E':(real^3->bool)->bool`]CARD_EQ_0)
THEN RESA_TAC
THEN EXISTS_TAC`a+ (&1-a:real) / &2`
THEN MP_TAC(REAL_ARITH`&0<a /\ a < &1 ==> a< a+ (&1-a:real) / &2/\ a+ (&1-a:real) / &2 <= &1`)
THEN RESA_TAC
THEN ASM_SET_TAC[];(*1*)

REPEAT GEN_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC "A")
THEN REPEAT STRIP_TAC
THEN MP_TAC(ISPEC`(E':(real^3->bool)->bool)` CHOOSE_SUBSET)
THEN RESA_TAC
THEN POP_ASSUM(fun th-> MP_TAC(ISPEC`n:num `th))
THEN REWRITE_TAC[ARITH_RULE `n:num <= SUC n`; HAS_SIZE]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE` t SUBSET E' /\ E' SUBSET E ==> (t:(real^3->bool)->bool) SUBSET E`)
THEN RESA_TAC
THEN REMOVE_THEN "A" (fun th-> MP_TAC(ISPEC`(t:(real^3->bool)->bool)`th))
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"YEU")
THEN SUBGOAL_THEN `~((E':(real^3->bool)->bool) DIFF (t:(real^3->bool)->bool)= {})` ASSUME_TAC
THENL(*2*)[
STRIP_TAC
THEN MP_TAC(SET_RULE`(E':(real^3->bool)->bool) DIFF (t:(real^3->bool)->bool)={} /\ t SUBSET E' ==>  t= E'`)
THEN RESA_TAC
THEN FIND_ASSUM MP_TAC`CARD (t:(real^3->bool)->bool)=n`
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN ASM_REWRITE_TAC[]
THEN ARITH_TAC;(*2*)
SUBGOAL_THEN`?e. e IN (E':(real^3->bool)->bool) DIFF (t:(real^3->bool)->bool)` ASSUME_TAC
THENL(*3*)[
ASM_SET_TAC[];(*3*)

POP_ASSUM MP_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`e IN (E':(real^3->bool)->bool) DIFF (t:(real^3->bool)->bool)/\
(E':(real^3->bool)->bool) SUBSET (E:(real^3->bool)->bool) /\ t SUBSET E' ==> e IN E'/\ e IN E/\ ~(e IN t) /\ {e} UNION t SUBSET E'`)
THEN RESA_TAC
THEN MP_TAC(ISPECL [`{e:(real^3->bool)} UNION (t:(real^3->bool)->bool)`;`(E':(real^3->bool)->bool)`] FINITE_SUBSET)
THEN RESA_TAC
THEN ASSUME_TAC(SET_RULE`e IN {e:(real^3->bool)} UNION (t:(real^3->bool)->bool)`)
THEN MP_TAC(ISPECL[`e:real^3->bool`;`{e:(real^3->bool)} UNION (t:(real^3->bool)->bool)`;]CARD_DELETE)
THEN RESA_TAC
THEN MP_TAC(SET_RULE `e IN {e:(real^3->bool)} UNION (t:(real^3->bool)->bool) 
==> ({e:(real^3->bool)} UNION (t:(real^3->bool)->bool)) DELETE e PSUBSET {e:(real^3->bool)} UNION (t:(real^3->bool)->bool)`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`({e:(real^3->bool)} UNION (t:(real^3->bool)->bool))DELETE e`;`{e:(real^3->bool)} UNION (t:(real^3->bool)->bool)`]CARD_PSUBSET)
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN FIND_ASSUM MP_TAC`FINITE ( {e:(real^3->bool)} UNION (t:(real^3->bool)->bool))`
THEN DISCH_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN DISCH_TAC
THEN MP_TAC(ARITH_RULE`CARD (({e:(real^3->bool)} UNION (t:(real^3->bool)->bool)) DELETE e) < CARD ( {e:(real^3->bool)} UNION (t:(real^3->bool)->bool))
/\ CARD (({e:(real^3->bool)} UNION (t:(real^3->bool)->bool)) DELETE e) = CARD ({e:(real^3->bool)} UNION (t:(real^3->bool)->bool))-1
<=>CARD (({e:(real^3->bool)} UNION (t:(real^3->bool)->bool)) DELETE e) +1= CARD ({e:(real^3->bool)} UNION (t:(real^3->bool)->bool))`)
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th;])
THEN REWRITE_TAC[ARITH_RULE`A=A`; ]
THEN MP_TAC(SET_RULE`~(e IN t)==>({e:(real^3->bool)} UNION (t:(real^3->bool)->bool)) DELETE e=t`)
THEN RESA_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN FIND_ASSUM MP_TAC`(CARD (E':(real^3->bool)->bool)=SUC n)`
THEN REWRITE_TAC[ARITH_RULE`SUC n=(n:num) +1`]
THEN DISCH_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)])
THEN DISCH_TAC
THEN MP_TAC(ISPECL[`{e:(real^3->bool)} UNION (t:(real^3->bool)->bool)`;`E':(real^3->bool)->bool`]CARD_SUBSET_EQ)
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)])
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"MA")
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool) `;`(E:(real^3->bool)->bool)`;`(e:real^3->bool)`]expand_edge_graph_fan)
THEN RESA_TAC
THEN MRESA_TAC fan_run_in_small_not0_is_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;` u:real^3`;`w:real^3`;`v':real^3`;`w':real^3`;`a:real`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th)] THEN ASSUME_TAC(th))
THEN POP_ASSUM MP_TAC
THEN RESA_TAC
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"BE")
THEN EXISTS_TAC` min (h:real) (t1:real)`
THEN STRIP_TAC
THENL(*4*)[
ASM_TAC
THEN REAL_ARITH_TAC;(*4*)

STRIP_TAC
THENL(*5*)[
ASM_TAC
THEN REAL_ARITH_TAC;(*5*)
REPEAT STRIP_TAC
THEN REMOVE_THEN "MA" MP_TAC
THEN DISCH_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th)] THEN ASSUME_TAC(th))
THEN REWRITE_TAC[UNION;IN_ELIM_THM;EXTENSION; INTER;]
THEN GEN_TAC
THEN EQ_TAC
THENL(*6*)[
ASM_REWRITE_TAC[IN_SING]
THEN MP_TAC(REAL_ARITH`s< min h t1==> s<h /\ s<t1`)
THEN RESA_TAC
THEN STRIP_TAC
THENL(*7*)[


POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN REMOVE_THEN "BE"(fun th-> MRESA1_TAC th `s:real`)
THEN ASM_SET_TAC[];

POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN REMOVE_THEN "YEU"(fun th-> MRESA1_TAC th `s:real`)
THEN ASM_SET_TAC[]];
SET_TAC[]]]]]]]);;






let cut_in_edges_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<a /\ a<= &1
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\  ~(aff_gt{x} {v,(&1-a)%u+ a % w} INTER {v | ?e. e IN E /\ v IN aff_ge {x} e}={})
==> a= &1 `,
REPEAT STRIP_TAC
THEN FIND_ASSUM MP_TAC `fan80(x:real^3,V,E)`
THEN REWRITE_TAC[fan80]
THEN DISCH_TAC
THEN POP_ASSUM (fun th -> MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC "YEU EM"))
THEN ABBREV_TAC`s1={h:real| &0< h/\ h<= &1 /\
~(aff_gt{x} {v,(&1-h)%u+ h % w:real^3} INTER {v | ?e. e IN E /\ v IN aff_ge {x:real^3} e}={})}`
THEN SUBGOAL_THEN`(a:real) IN (s1:real->bool)` ASSUME_TAC
THENL(*1*)[
POP_ASSUM(fun th->ASM_REWRITE_TAC[SYM(th);IN_ELIM_THM]);

MP_TAC(SET_RULE`(a:real) IN (s1:real->bool)==> ~(s1={})`)
THEN RESA_TAC
THEN SUBGOAL_THEN`(?b. !x:real. x IN s1 ==> b <= x)`ASSUME_TAC
THENL(*2*)[
EXISTS_TAC`&0`
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th->ASM_REWRITE_TAC[SYM(th);IN_ELIM_THM])
THEN REAL_ARITH_TAC;
MRESA1_TAC INF`s1:real->bool`
THEN ABBREV_TAC`b= inf (s1:real->bool)`
THEN SUBGOAL_THEN`b<= &1` ASSUME_TAC
THENL(*3*)[POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th-> MRESA1_TAC th `a:real`)
THEN ASM_TAC
THEN REAL_ARITH_TAC;(*3*)
 
SUBGOAL_THEN`~(?s. &0 < s /\ s < b
          /\ ~( aff_gt {x} {v, (&1 - s) % u + s % w} INTER
              {v | ?e. e IN E /\ v IN aff_ge {x:real^3} e} =
              {}))` ASSUME_TAC
THENL(*4*)[STRIP_TAC
THEN SUBGOAL_THEN`s IN (s1:real->bool)` ASSUME_TAC
THENL(*5*)[ EXPAND_TAC"s1"
THEN REWRITE_TAC[IN_ELIM_THM]
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC;(*5*)
POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN  POP_ASSUM (fun th->MP_TAC(ISPEC`s:real` th))
THEN DISCH_THEN (LABEL_TAC"DICH")
THEN REPEAT STRIP_TAC
THEN REMOVE_THEN"DICH" MP_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC];(*4*)
POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM NOT_IMP; GSYM NOT_FORALL_THM;IMP_IMP]
THEN STRIP_TAC
THEN MRESA_TAC fan_run_in_small_is_not_meet_xfan [`x:real^3`;` (V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` v:real^3`;`u:real^3`;`w:real^3`] 
THEN SUBGOAL_THEN`~(?x:real. x IN s1 /\ ~(h <= x))`ASSUME_TAC
THENL(*5*)[
POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"A")
THEN EXPAND_TAC"s1"
THEN REWRITE_TAC[IN_ELIM_THM; REAL_ARITH`~(A<=B)<=> B<A`]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN REMOVE_THEN"A"(fun th-> MRESA1_TAC th `x':real`);(*5*)

POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM NOT_IMP; GSYM NOT_FORALL_THM]
THEN STRIP_TAC
THEN SUBGOAL_THEN`&0<b:real`ASSUME_TAC
THENL(*6*)[


POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN  POP_ASSUM (fun th->MP_TAC(ISPEC`h:real` th))
THEN DISCH_THEN (LABEL_TAC"DICH")
THEN REPEAT STRIP_TAC
THEN REMOVE_THEN"DICH" MP_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;(*6*)
MRESA_TAC not_cut_inside_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3` ;`b:real`]
THEN POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(b:real)< &1\/ &1 <= b`)
THENL(*7*)[
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MRESAL_TAC fan_run1_in_small_not0_is_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`b:real`][SET_RULE`A SUBSET A`]
THEN SUBGOAL_THEN`~(?x:real. x IN s1 /\ ~(h' <= x))`ASSUME_TAC
THENL(*8*)[
POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"A")
THEN EXPAND_TAC"s1"
THEN REWRITE_TAC[IN_ELIM_THM; REAL_ARITH`~(A<=B)<=> B<A`]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN REMOVE_THEN"A"(fun th-> MRESA1_TAC th `x':real`);(*8*)
POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM NOT_IMP; GSYM NOT_FORALL_THM]
THEN STRIP_TAC
THEN SUBGOAL_THEN `h':real <= b`ASSUME_TAC
THENL(*9*)[POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN  POP_ASSUM (fun th->MP_TAC(ISPEC`h':real` th))
THEN REWRITE_TAC[];(*9*)
POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC](*9*)](*8*);(*7*)

STRIP_TAC
THEN MP_TAC(REAL_ARITH`&1<=b /\ b<= &1==> b= &1`) THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN REMOVE_ASSUM_TAC   
THEN  POP_ASSUM (fun th->MP_TAC(ISPEC`a:real` th))
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC]]]]]]]);;





let not_cut_in_edges_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<a /\ a< &1
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
==>  (aff_gt{x} {v,(&1-a)%u+ a % w} INTER {v | ?e. e IN E /\ v IN aff_ge {x} e}={})`,

REPEAT STRIP_TAC
THEN DISJ_CASES_TAC(SET_RULE`(?a. &0< a /\ a< &1 /\ ~(aff_gt{x} {v,(&1-a)%u+ a % w} INTER {v | ?e. e IN E /\ v IN aff_ge {x} e}={})) \/  (!a. &0< a /\ a< &1 ==> (aff_gt{x} {v,(&1-a)%u+ a % w} INTER {v | ?e. e IN E /\ v IN aff_ge {x:real^3} e}={})) `)
THENL[
POP_ASSUM MP_TAC
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"EM BE")
THEN FIND_ASSUM MP_TAC `fan80(x:real^3,V,E)`
THEN REWRITE_TAC[fan80]
THEN DISCH_TAC
THEN POP_ASSUM (fun th -> MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC "YEU EM"))
THEN ABBREV_TAC`s1={h:real| &0< h/\ h< &1 /\
~(aff_gt{x} {v,(&1-h)%u+ h % w:real^3} INTER {v | ?e. e IN E /\ v IN aff_ge {x:real^3} e}={})}`
THEN SUBGOAL_THEN`(a':real) IN (s1:real->bool)` ASSUME_TAC
THENL(*1*)[
POP_ASSUM(fun th->ASM_REWRITE_TAC[SYM(th);IN_ELIM_THM]);
MP_TAC(SET_RULE`(a':real) IN (s1:real->bool)==> ~(s1={})`)
THEN RESA_TAC
THEN SUBGOAL_THEN`(?b. !x:real. x IN s1 ==> b <= x)`ASSUME_TAC
THENL(*2*)[EXISTS_TAC`&0`
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th->ASM_REWRITE_TAC[SYM(th);IN_ELIM_THM])
THEN REAL_ARITH_TAC;(*2*)
MRESA1_TAC INF`s1:real->bool`
THEN ABBREV_TAC`b= inf (s1:real->bool)`
THEN SUBGOAL_THEN`b< &1` ASSUME_TAC
THENL(*3*)[
POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th-> MRESA1_TAC th `a':real`)
THEN ASM_TAC
THEN REAL_ARITH_TAC;(*3*)
 SUBGOAL_THEN`~(?s. &0 < s /\ s < b
          /\ ~( aff_gt {x} {v, (&1 - s) % u + s % w} INTER
              {v | ?e. e IN E /\ v IN aff_ge {x:real^3} e} =
              {}))` ASSUME_TAC
THENL(*4*)[
STRIP_TAC
THEN SUBGOAL_THEN`s IN (s1:real->bool)` ASSUME_TAC
THENL(*5*)[EXPAND_TAC"s1"
THEN REWRITE_TAC[IN_ELIM_THM]
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC;(*5*)
POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN  POP_ASSUM (fun th->MP_TAC(ISPEC`s:real` th))
THEN DISCH_THEN (LABEL_TAC"DICH")
THEN REPEAT STRIP_TAC
THEN REMOVE_THEN"DICH" MP_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC];(*4*)

POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM NOT_IMP; GSYM NOT_FORALL_THM;IMP_IMP]
THEN STRIP_TAC
THEN MRESA_TAC fan_run_in_small_is_not_meet_xfan [`x:real^3`;` (V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` v:real^3`;`u:real^3`;`w:real^3`] 
THEN SUBGOAL_THEN`~(?x:real. x IN s1 /\ ~(h <= x))`ASSUME_TAC
THENL(*5*)[POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"A")
THEN EXPAND_TAC"s1"
THEN REWRITE_TAC[IN_ELIM_THM; REAL_ARITH`~(A<=B)<=> B<A`]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN REMOVE_THEN"A"(fun th-> MRESA1_TAC th `x':real`);(*5*)

POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM NOT_IMP; GSYM NOT_FORALL_THM]
THEN STRIP_TAC
THEN SUBGOAL_THEN`&0<b:real`ASSUME_TAC
THENL(*6*)[
POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN  POP_ASSUM (fun th->MP_TAC(ISPEC`h:real` th))
THEN DISCH_THEN (LABEL_TAC"DICH")
THEN REPEAT STRIP_TAC
THEN REMOVE_THEN"DICH" MP_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;(*6*)
MRESA_TAC not_cut_inside_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3` ;`b:real`]
THEN MRESAL_TAC fan_run1_in_small_not0_is_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`b:real`][SET_RULE`A SUBSET A`]
THEN SUBGOAL_THEN`~(?x:real. x IN s1 /\ ~(h' <= x))`ASSUME_TAC
THENL(*7*)[
POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"A")
THEN EXPAND_TAC"s1"
THEN REWRITE_TAC[IN_ELIM_THM; REAL_ARITH`~(A<=B)<=> B<A`]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN REMOVE_THEN"A"(fun th-> MRESA1_TAC th `x':real`);(*7*)
POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM NOT_IMP; GSYM NOT_FORALL_THM]
THEN STRIP_TAC
THEN SUBGOAL_THEN `h':real <= b`ASSUME_TAC
THENL(*8*)[POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC  
THEN  POP_ASSUM (fun th->MP_TAC(ISPEC`h':real` th))
THEN REWRITE_TAC[];

POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC]]]]]]]];

POP_ASSUM (fun th-> MRESA1_TAC th`a:real`)]);;




let lie_in_half_space_and_azim_le=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real v1:real^3 u1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<a /\ a<= &1
/\ fan80(x,V,E)
/\ ~collinear{x,v1,u1}
/\ v1 IN aff_gt {x} {v,(&1-a)%u+ a%w}
/\ &0< azim x v1 v u1
/\  azim x v1 v u1 < pi
==> &0 < ((v-x) cross (u - x)) dot (v1-x)`,


REPEAT STRIP_TAC
THEN FIND_ASSUM MP_TAC `fan80(x:real^3,V,E)`
THEN REWRITE_TAC[fan80]
THEN DISCH_THEN(LABEL_TAC"yeu em")
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (u:real^3)`;`v:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] 
THEN REMOVE_THEN "yeu em" (fun th -> MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC "YEU EM"))
THEN MP_TAC(ARITH_RULE`&0<a==> &0<= a`) THEN RESA_TAC
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded1 [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - a) % (u:real^3) + a % (w:real^3)`;]
THEN MRESA_TAC AFF_GT_1_2[`(x:real^3)` ;` (v:real^3)`;`(&1 - a) % (u:real^3) + a % (w:real^3) `;]
THEN FIND_ASSUM MP_TAC `v1 IN aff_gt {x} {v,(&1-a)%u+ a%(w:real^3)}`
THEN POP_ASSUM(fun th-> REWRITE_TAC[th;IN_ELIM_THM])
THEN STRIP_TAC 
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A- vec 0= A`;VECTOR_ARITH`(t1 % x + t2 % v + t3 % ((&1 - a) % u + a % w)) - x=(t1- &1) % x + t2 % v + t3 % ((&1 - a) % u + a % w)`]
THEN FIND_ASSUM MP_TAC `t1 + t2 + t3 = &1:real`
THEN REWRITE_TAC[REAL_ARITH`t1 + t2 + t3 = &1<=> t1- &1= (-- t2)- t3 `]
THEN DISCH_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(--t2 - t3) % x + t2 % v + t3 % ((&1 - a) % u + a % w)= t2 % (v-x) + t3 % ((&1 - a) % (u-x) + a % (w-x))`]
THEN REWRITE_TAC[ DOT_RADD;DOT_RMUL;DOT_CROSS_SELF]
THEN REDUCE_ARITH_TAC
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`; `u:real^3`;`v:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[CROSS_TRIPLE]);;

let cross_dot_fully_surrounded1_fan_le=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real v1:real^3 u1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<a /\ a<= &1
/\ fan80(x,V,E)
/\ ~collinear{x,v1,u1}
/\ v1 IN aff_gt {x} {v,(&1-a)%u+ a%w}
/\ &0< azim x v1 v u1
/\  azim x v1 v u1 < pi
==> &0< ((v1 - x) cross (u1 - x)) dot
  ((&1 - a) % u + a % w-x)`,

REPEAT STRIP_TAC
THEN MRESAL_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`v1:real^3`; `(&1 - a) % u + a % w:real^3`;`u1:real^3`][VECTOR_ARITH`((&1 - a) % u + a % w) - x=(&1 - a) % u + a % w - x`]
THEN POP_ASSUM MATCH_MP_TAC
THEN FIND_ASSUM MP_TAC `fan80(x:real^3,V,E)`
THEN REWRITE_TAC[fan80]
THEN DISCH_THEN(LABEL_TAC"yeu em")
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (u:real^3)`;`v:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] 
THEN REMOVE_THEN "yeu em" (fun th -> MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC "YEU EM"))
THEN MP_TAC(ARITH_RULE`&0<a==> &0<= a`) THEN RESA_TAC
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded1 [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`]
THEN MRESA_TAC properties_of_collinear4_points_fan[`x:real^3`;`v:real^3`;`(&1-a)%u+ a%w:real^3`;`v1:real^3`]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - a) % (u:real^3) + a % (w:real^3)`;]
THEN MRESA_TAC AFF_GT_1_2[`(x:real^3)` ;` (v:real^3)`;`(&1 - a) % (u:real^3) + a % (w:real^3) `;]
THEN FIND_ASSUM MP_TAC `(v1:real^3) IN aff_gt {x} {v,(&1-a)%u+ a%(w:real^3)}`
THEN POP_ASSUM(fun th-> REWRITE_TAC[th;IN_ELIM_THM])
THEN STRIP_TAC 
THEN SUBGOAL_THEN`~collinear {x, v1, (&1 - a) % u + a % w:real^3}` ASSUME_TAC
THENL[
ASM_REWRITE_TAC[collinear1_fan;]
THEN FIND_ASSUM MP_TAC`~((&1 - a) % u + a % w IN aff {x, v:real^3})`
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1 % x + t2 % v + t3 % ((&1 - a) % u + a % w) =
      u' % x + v' % ((&1 - a) % u + a % w)<=>
(&1-(t1+t2+t3)) % x  + (v'-t3) % ((&1 - a) % u + a % w-x) =
      (&1-(u'+v') ) % x + t2 % (v-x)`;REAL_ARITH`a-a= &0`]
THEN REDUCE_VECTOR_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`v' - t3= &0 \/ ~(v' - (t3:real)= &0)`)
THENL[
ASM_REWRITE_TAC[VECTOR_ARITH`&0 %A=B<=> B= vec 0`;VECTOR_MUL_EQ_0]
THEN MP_TAC(REAL_ARITH`&0< t2==> ~(t2 = &0)`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A-B= vec 0<=> A=B`];

MRESA1_TAC REAL_MUL_LINV `(v'-t3:real)`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(v' - t3) % ((&1 - a) % u + a % w - x) = t2 % (v - x:real^3) ==> (inv (v'-t3)) % (v' - t3) % ((&1 - a) % u + a % w - x) = (inv (v'-t3)) % ( t2 % (v - x))`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th] 
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(A%B%C=(A*B)%C:real^3)`])
THEN REWRITE_TAC[VECTOR_ARITH`&1 % ((&1 - a) % u + a % w - x) = (inv (v' - t3) * t2) % (v - x)<=>
(&1 - a) % u + a % w  = (&1-(inv (v' - t3) * t2))%x +(inv (v' - t3) * t2) % v:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`&1-(inv (v' - t3) * (t2:real))`
THEN EXISTS_TAC`(inv (v' - t3) * (t2:real))`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC];

POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"A")
THEN DISCH_TAC
THEN REMOVE_THEN "A" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`v1:real^3`;`v:real^3`;`u1:real^3`;`((&1 - a) % u + a % w):real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"B")
THEN DISCH_TAC
THEN SUBGOAL_THEN`azim x v1 v ((&1 - a) % u + a % w)= pi` ASSUME_TAC
THENL[
POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`(x:real^3)` ;`v1:real^3`;` (v:real^3)`]
THEN MRESA_TAC AFF_LT_2_1[`x:real^3`;`v1:real^3`;`v:real^3`]
THEN MRESAL_TAC AZIM_EQ_PI_ALT[`x:real^3`;`v1:real^3`;`v:real^3`;`((&1 - a) % u + a % w):real^3`][IN_ELIM_THM]
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[VECTOR_ARITH`A=B+C+D <=> D=A-B-C:real^3`]
THEN MP_TAC(REAL_ARITH`&0<t3==> ~(t3= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV `(t3:real)`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`t3 % ((&1 - a) % u + a % w) = v1 - t1 % x - t2 % v ==> (inv (t3)) % t3 % ((&1 - a) % u + a % w)  = (inv (t3)) % ( v1 - t1 % x - t2 % v:real^3)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th] 
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(A%B%C=(A*B)%C:real^3)`])
THEN REWRITE_TAC[VECTOR_ARITH`&1 % ((&1 - a) % u + a % w) = inv t3 % (v1 - t1 % x - t2 % v)
<=> (&1 - a) % u + a % w = (-- inv t3 * t1) % x + (inv t3) % v1 +(-- (inv t3* t2)) % v`]
THEN STRIP_TAC
THEN EXISTS_TAC`(-- inv t3 * t1):real`
THEN EXISTS_TAC`(inv t3):real`
THEN EXISTS_TAC`(-- (inv t3 * t2)):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`--inv t3 * t1 + inv t3 + --(inv t3 * t2)= inv t3 *(t3+ &1-(t1+t2+t3))`;REAL_ARITH`(t3 + &1 - &1)=t3`;REAL_ARITH`--A< &0 <=> &0< A`]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LT_INV
THEN ASM_REWRITE_TAC[];
REMOVE_THEN "B" MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN REMOVE_ASSUM_TAC
THEN MP_TAC(REAL_ARITH`azim x v1 v u1 < pi==> azim x v1 v u1 <= pi`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN FIND_ASSUM MP_TAC`azim x v1 v (u1:real^3) < pi`
THEN FIND_ASSUM MP_TAC`&0< azim x v1 v (u1:real^3) `
THEN REAL_ARITH_TAC]]);;




let exists_cross_dot_fully_surrounded1_fan_le=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real v1:real^3 u1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<a /\ a<= &1
/\ fan80(x,V,E)
/\ ~collinear{x,v1,u1}
/\ v1 IN aff_gt {x} {v,(&1-a)%u+ a%w}
/\ &0< azim x v1 v u1
/\  azim x v1 v u1 < pi
==> ?t.  &0< t/\ t < &1
/\( !h:real. &0< h/\ h<t
==> &0< ((v1 - x) cross (u1 - x)) dot (((&1 - h) % ((&1 - a) % u + a % w) + h % u) - x))`,

REPEAT STRIP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`(((&1 - t) % ((&1 - a) % u + a % w) + t % u) - x)=
 ((&1 - a) % u + a % w - x) + t % (--a) % ( w-u)`;]
THEN ABBREV_TAC`va=(&1 - a) % u + a % w-x:real^3`
THEN REWRITE_TAC[DOT_RADD;DOT_RMUL; REAL_ARITH`&0<A+C*(--B)*D<=> (B*D)*C<A`]
THEN MRESA_TAC cross_dot_fully_surrounded1_fan_le[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` v:real^3`;` u:real^3`;`w:real^3`;`a:real`;`v1:real^3`;`u1:real^3`]
THEN DISJ_CASES_TAC(REAL_ARITH`(((v1 - x) cross (u1 - x:real^3)) dot (w - u)<= &0 )\/ (&0<((v1 - x) cross (u1 - x)) dot (w - u))`)
THENL[ 
EXISTS_TAC`&1/ &2`
THEN REWRITE_TAC[REAL_ARITH`&0 < &1 / &2 /\ &1 / &2 < &1`]
THEN REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0 < a ==> &0<= (a:real)`) THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0 < h ==> &0<= (h:real)`) THEN RESA_TAC
THEN MRESAL_TAC REAL_LE_LMUL[`a:real`;`((v1 - x) cross (u1 - x)) dot ((w - u):real^3)`;`&0`][REAL_ARITH`(A:real)*  &0= &0`]
THEN MRESAL_TAC REAL_LE_RMUL[`a*((v1 - x) cross (u1 - x)) dot ((w - u):real^3)`;`&0`;`h:real`][REAL_ARITH`&0 * (A:real)= &0`]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MRESA_TAC REAL_LT_MUL[`a:real`;`((v1 - x) cross (u1 - x)) dot ((w - u):real^3)`]
THEN MP_TAC(REAL_ARITH`&0<(a:real)*((v1 - x) cross (u1 - x)) dot ((w - u):real^3)==> ~( (a:real)*((v1 - x) cross (u1 - x)) dot ((w - u):real^3)= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_RINV`(a:real)*((v1 - x) cross (u1 - x)) dot ((w - u):real^3)`
THEN MRESA1_TAC REAL_LT_INV`(a:real)*((v1 - x) cross (u1 - x)) dot ((w - u):real^3)`
THEN MRESA_TAC REAL_LT_MUL[`inv((a:real)*((v1 - x) cross (u1 - x)) dot ((w - u):real^3))`;`((v1 - x) cross (u1 - x)) dot (va:real^3)`]
THEN EXISTS_TAC `min ((inv(a *(((v1 - x) cross (u1 - x)) dot (w - u)))*(((v1 - x) cross (u1 - x)) dot va)) / &2 ) (&1/ &2)`
THEN STRIP_TAC
THENL[POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
STRIP_TAC
THENL[
 REAL_ARITH_TAC;
REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH` h< min ((inv (a * (((v1 - x) cross (u1 - x)) dot (w - u))) * (((v1 - x) cross (u1 - x)) dot va)) /  &2) (&1 / &2)
==>
h< ((inv (a * (((v1 - x) cross (u1 - x)) dot (w - u))) * (((v1 - x) cross (u1 - x)) dot va)) /  &2)`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_LMUL[`(a * (((v1 - x) cross (u1 - x)) dot (w - u:real^3)))`;`h:real`;`((inv (a * (((v1 - x) cross (u1 - x)) dot (w - u))) * (((v1 - x) cross (u1 - x:real^3)) dot va)) /  &2)`]
THEN ASM_TAC
THEN ABBREV_TAC`vb=a * (((v1 - x) cross (u1 - x)) dot (w - u))`
THEN REPEAT DISCH_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`vb * (inv vb * (((v1 - x) cross (u1 - x)) dot va)) / &2=
(vb * inv vb) * (((v1 - x) cross (u1 - x)) dot va)/ &2`]
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC]]]);;

let cross_dot_fully_surrounded2_fan_le=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real v1:real^3 u1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<a /\ a<= &1
/\ fan80(x,V,E)
/\ ~collinear{x,v1,u1}
/\ v1 IN aff_gt {x} {v,(&1-a)%u+ a%w}
/\ &0< azim x v1 v u1
/\  azim x v1 v u1 < pi
==> &0 < (((&1 - a) % u + a % w - x) cross (v - x)) dot (u1 - x)`,

REPEAT STRIP_TAC
THEN FIND_ASSUM MP_TAC `fan80(x:real^3,V,E)`
THEN REWRITE_TAC[fan80]
THEN DISCH_THEN(LABEL_TAC"yeu em")
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (u:real^3)`;`v:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] 
THEN REMOVE_THEN "yeu em" (fun th -> MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC "YEU EM"))
THEN MP_TAC(ARITH_RULE`&0<a==> &0<= a`) THEN RESA_TAC
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded1 [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`]
THEN MRESA_TAC properties_of_collinear4_points_fan[`x:real^3`;`v:real^3`;`(&1-a)%u+ a%w:real^3`;`v1:real^3`]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - a) % (u:real^3) + a % (w:real^3)`;]
THEN MRESA1_TAC SIN_POS_PI`azim x v1 v (u1:real^3)`
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC AZIM_TRANSLATION[`-- x:real^3`;`x:real^3`;`v1:real^3`;` v:real^3`;`u1:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`x-x= vec 0`;VECTOR_ARITH`(-- X)+A= A-X:real^3`]
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN MRESA_TAC JBDNJJB[`(v1-x):real^3`;`v-x:real^3`;`u1-x:real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM COLLINEAR_3;]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN RESA_TAC
THEN MRESAL_TAC REAL_LT_LMUL_EQ [` &0:real `;`(((v1 - x) cross (v - x)) dot ((u1 - x):real^3)):real`;`t:real`][REAL_ARITH`a * &0 = &0`]
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN MRESA_TAC AFF_GT_1_2[`(x:real^3)` ;` (v:real^3)`;`(&1 - a) % (u:real^3) + a % (w:real^3) `;]
THEN FIND_ASSUM MP_TAC `v1 IN aff_gt {x} {v,(&1-a)%u+ a%(w:real^3)}`
THEN POP_ASSUM(fun th-> REWRITE_TAC[th;IN_ELIM_THM])
THEN STRIP_TAC 
THEN POP_ASSUM (fun th-> REWRITE_TAC[th;VECTOR_ARITH`A- vec 0= A`;VECTOR_ARITH`(t1 % x + t2 % v + t3 % ((&1 - a) % u + a % w)) - x=(t1- &1) % x + t2 % v + t3 % ((&1 - a) % u + a % w)`])
THEN FIND_ASSUM MP_TAC `t1 + t2 + t3 = &1:real`
THEN REWRITE_TAC[REAL_ARITH`t1 + t2 + t3 = &1<=> t1- &1= (-- t2)- t3 `]
THEN DISCH_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(--t2 - t3) % x + t2 % v + t3 % ((&1 - a) % u + a % w)= t2 % (v-x) + t3 % ((&1 - a) % u + a % w-x)`]
THEN REWRITE_TAC[CROSS_LADD; CROSS_RADD; CROSS_LMUL; CROSS_RMUL;CROSS_REFL;CROSS_RNEG;CROSS_LNEG]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[GSYM CROSS_LMUL;GSYM CROSS_LADD;DOT_LMUL]
THEN MRESAL_TAC REAL_LT_LMUL_EQ [` &0:real `;`((((&1 - a) % u + a % w - x) cross (v - x)) dot (u1 - x)):real`;`t3:real`][REAL_ARITH`a * &0 = &0`]);;


let exists_cross_dot_fully_surrounded2_fan_le=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real v1:real^3 u1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<a /\ a<= &1
/\ fan80(x,V,E)
/\ ~collinear{x,v1,u1}
/\ v1 IN aff_gt {x} {v,(&1-a)%u+ a%w}
/\ &0< azim x v1 v u1
/\  azim x v1 v u1 < pi
==>
?t:real.  &0< t/\ t < &1
/\ (!h:real. &0<h /\ h< t 
==> &0 <
((((&1 - h) % ((&1 - a) % u + a % w) + h % u) - x) cross (v - x)) dot (u1 - x))`,

REPEAT STRIP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`(((&1 - t) % ((&1 - a) % u + a % w) + t % u) - x)=
 ((&1 - a) % u + a % w - x) + t % (--a) % ( w-u)`;]
THEN ABBREV_TAC`va=(&1 - a) % u + a % w-x:real^3`
THEN REWRITE_TAC[CROSS_LADD;CROSS_LMUL;DOT_LADD;DOT_LMUL; REAL_ARITH`&0<A+C*(--B)*D<=> (B*D)*C<A`]
THEN MRESA_TAC cross_dot_fully_surrounded2_fan_le[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` v:real^3`;` u:real^3`;`w:real^3`;`a:real`;`v1:real^3`;`u1:real^3`]
THEN DISJ_CASES_TAC(REAL_ARITH`(((w - u) cross (v - x:real^3)) dot (u1- x)<= &0 )\/ (&0<((w - u) cross (v - x:real^3)) dot (u1- x))`)
THENL[ 
EXISTS_TAC`&1/ &2`
THEN REWRITE_TAC[REAL_ARITH`&0 < &1 / &2 /\ &1 / &2 < &1`]
THEN REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0 < a ==> &0<= (a:real)`) THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0 < h ==> &0<= (h:real)`) THEN RESA_TAC
THEN MRESAL_TAC REAL_LE_LMUL[`a:real`;`((w - u) cross (v - x:real^3)) dot (u1- x:real^3)`;`&0`][REAL_ARITH`(A:real)*  &0= &0`]
THEN MRESAL_TAC REAL_LE_RMUL[`a*(((w - u) cross (v - x:real^3)) dot (u1- x))`;`&0`;`h:real`][REAL_ARITH`&0 * (A:real)= &0`]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MRESA_TAC REAL_LT_MUL[`a:real`;`((w - u) cross (v - x:real^3)) dot (u1- x:real^3)`]
THEN MP_TAC(REAL_ARITH`&0<(a:real)*(((w - u) cross (v - x:real^3)) dot (u1- x))==> ~( (a:real)*(((w - u) cross (v - x:real^3)) dot (u1- x))= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_RINV`(a:real)*(((w - u) cross (v - x:real^3)) dot (u1- x))`
THEN MRESA1_TAC REAL_LT_INV`(a:real)*(((w - u) cross (v - x:real^3)) dot (u1- x))`
THEN MRESA_TAC REAL_LT_MUL[`inv((a:real)*(((w - u) cross (v - x:real^3)) dot (u1- x)))`;`((va cross (v - x)) dot (u1 - x))`]
THEN EXISTS_TAC `min ((inv(a *(((w - u) cross (v - x:real^3)) dot (u1- x)))*((va cross (v - x)) dot (u1 - x))) / &2 ) (&1/ &2)`
THEN STRIP_TAC
THENL[POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
STRIP_TAC
THENL[
 REAL_ARITH_TAC;
REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH` h< min ((inv(a *(((w - u) cross (v - x:real^3)) dot (u1- x)))*((va cross (v - x)) dot (u1 - x))) / &2 ) (&1/ &2)
==>
h< ((inv(a *(((w - u) cross (v - x:real^3)) dot (u1- x)))*((va cross (v - x)) dot (u1 - x))) / &2 )`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_LMUL[`(a *(((w - u) cross (v - x:real^3)) dot (u1- x)))`;`h:real`;`((inv(a *(((w - u) cross (v - x:real^3)) dot (u1- x)))*((va cross (v - x)) dot (u1 - x))) / &2 )`]
THEN ASM_TAC
THEN ABBREV_TAC`vb=a * (((w - u) cross (v - x:real^3)) dot (u1- x))`
THEN REPEAT DISCH_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`vb * (inv vb * ((va cross (v - x)) dot (u1 - x))) / &2=
(vb * inv vb) * ((va cross (v - x)) dot (u1 - x))/ &2`]
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC]]]);;







let exists_cut_small_edges_fan_le=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real v1:real^3 u1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<a /\ a<= &1
/\ fan80(x,V,E)
/\ ~collinear{x,v1,u1}
/\ v1 IN aff_gt {x} {v,(&1-a)%u+ a%w}
/\ &0< azim x v1 v u1
/\  azim x v1 v u1 < pi
==>
?t:real.  &0< t/\ t < &1
/\ ~(aff_gt {x} {v,(&1-t)%((&1-a)%u+ a%w)+t%u} INTER aff_gt {x} {v1,u1}={})`,

REPEAT STRIP_TAC
THEN MRESA_TAC lie_in_half_space_and_azim_le[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`;`v1:real^3`;` u1:real^3`]
THEN MRESA_TAC exists_cross_dot_fully_surrounded1_fan_le[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`;`v1:real^3`;` u1:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"MA")
THEN MRESA_TAC exists_cross_dot_fully_surrounded2_fan_le[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`;`v1:real^3`;` u1:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"BE NHO")
THEN FIND_ASSUM MP_TAC `fan80(x:real^3,V,E)`
THEN REWRITE_TAC[fan80]
THEN DISCH_THEN(LABEL_TAC"yeu em")
THEN ABBREV_TAC`ta=min (t:real) (t':real)/ &2`
THEN EXISTS_TAC `(ta:real)`
THEN SUBGOAL_THEN `&0< ta:real /\ ta < &1 /\ ta< t /\ ta< t'` ASSUME_TAC
THENL(*1*)[
POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN FIND_ASSUM MP_TAC`&0<t:real`
THEN FIND_ASSUM MP_TAC`t:real < &1`
THEN FIND_ASSUM MP_TAC`&0<t':real`
THEN FIND_ASSUM MP_TAC`t':real < &1`
THEN REAL_ARITH_TAC;(*1*)

ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`&0<a==> &0<= a`) THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`ta< &1 ==> &0<= (&1-ta):real`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LE_MUL [`(&1-ta):real`;`a:real`]
THEN MRESA_TAC REAL_LE_LMUL [`(&1-ta):real`;`a:real`;`&1`]
THEN MP_TAC(REAL_ARITH` &0< ta /\ (&1-ta)*a<= (&1-ta)* &1 :real==> (&1-ta)*a<= &1`)
THEN RESA_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (u:real^3)`;`v:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`] 
THEN REMOVE_THEN "yeu em" (fun th -> MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC "YEU EM"))
THEN MRESAL_TAC not_collinear_is_properties_fully_surrounded1 [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`((&1-ta)*a):real`][VECTOR_ARITH`(&1 - ((&1 - ta) * a)) % u + ((&1 - ta) * a ) % w=
(&1 - ta) % ((&1 - a) % u + a % w) + ta % u:real^3`]
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded1 [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`]
THEN MRESA_TAC properties_of_collinear4_points_fan[`x:real^3`;`v:real^3`;`(&1-a)%u+ a%w:real^3`;`v1:real^3`]
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`v1:real^3`;`u1:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"CON")
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - ta) % ((&1 - a) % u + a % w) + ta % u:real^3`;]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v1:real^3)`;`u1:real^3`;]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - a) % (u:real^3) + a % (w:real^3)`;]
THEN MRESA_TAC AFF_GT_1_2[`(x:real^3)` ;` (v1:real^3)`;`(u1:real^3) `;]
THEN MRESA_TAC AFF_GT_1_2[`(x:real^3)` ;` (v:real^3)`;`(&1 - ta) % ((&1 - a) % u + a % w) + ta % u:real^3 `;]
THEN ASM_REWRITE_TAC[INTER;IN_ELIM_THM;EXTENSION;EMPTY;IN;NOT_FORALL_THM]
THEN ABBREV_TAC`a1=(v-x):real^3`
THEN ABBREV_TAC`a2=(((&1-ta)%((&1-a)%u+ a%w)+ta%u)-x):real^3`
THEN ABBREV_TAC`a3=(v1-x) :real^3`
THEN ABBREV_TAC`a4=u1-x:real^3`
THEN ABBREV_TAC`va=a1 cross a2:real^3`
THEN ABBREV_TAC`vb=a3 cross a4:real^3`
THEN EXISTS_TAC`(vb:real^3) cross (va:real^3)+(x:real^3)`
THEN STRIP_TAC
THENL(*2*)[
EXISTS_TAC`&1-(vb:real^3) dot (a2:real^3)+ vb dot (a1:real^3)`
THEN EXISTS_TAC`(vb:real^3) dot (a2:real^3)`
THEN EXISTS_TAC`--((vb:real^3) dot (a1:real^3))`
THEN REMOVE_THEN "MA"(fun th->  MRESA1_TAC th `ta:real`)
THEN ASM_REWRITE_TAC[REAL_ARITH`(&1 - vb dot a2 + vb dot a1) + vb dot a2 + --(vb dot a1) = &1`]
THEN SUBGOAL_THEN `&0< --((vb:real^3) dot (a1:real^3))` ASSUME_TAC
THENL(*3*)[
EXPAND_TAC"vb"
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[DOT_LNEG]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN REMOVE_THEN "CON"MP_TAC
THEN REAL_ARITH_TAC;(*3*)

ASM_REWRITE_TAC[]
THEN EXPAND_TAC"va"
THEN REWRITE_TAC[CROSS_LAGRANGE;VECTOR_MUL_LNEG]
THEN EXPAND_TAC"a1"
THEN EXPAND_TAC"a2"
THEN VECTOR_ARITH_TAC];(*2*)

ONCE_REWRITE_TAC[CROSS_SKEW]
THEN EXPAND_TAC"vb"
THEN REWRITE_TAC[CROSS_LAGRANGE;]
THEN EXISTS_TAC`&1+(va:real^3) dot (a4:real^3)- va dot (a3:real^3)`
THEN EXISTS_TAC`--(va:real^3) dot (a4:real^3)`
THEN EXISTS_TAC`((va:real^3) dot (a3:real^3))`
THEN REMOVE_THEN "BE NHO"(fun th->  MRESA1_TAC th `ta:real`)
THEN ASM_REWRITE_TAC[DOT_LNEG;VECTOR_MUL_LNEG;REAL_ARITH`(&1 + va dot a4 - va dot a3) + --(va dot a4) + va dot a3 = &1`;]
THEN STRIP_TAC
THENL(*3*)[
EXPAND_TAC"va"
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[DOT_LNEG]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;(*3*)

STRIP_TAC 
THENL(*4*)[
EXPAND_TAC"va"
THEN EXPAND_TAC"a2"
THEN REWRITE_TAC[VECTOR_ARITH`((&1 - ta) % ((&1 - a) % u + a % w) + ta % u) - x
=(&1 - ta) % ((&1 - a) % u + a % w-x) + ta % (u - x)`]
THEN ABBREV_TAC`vu=(&1 - a) % u + a % w - x:real^3`
THEN REWRITE_TAC[CROSS_RADD;CROSS_RMUL;DOT_LADD;DOT_LMUL;]
THEN MRESAL_TAC coplanar_is_cross_fan[`x:real^3`;`v:real^3`;`(&1 - a) % u + a % w:real^3`;`v1:real^3`]
[VECTOR_ARITH`((&1 - a) % u + a % w) - x=(&1 - a) % u + a % w - x`]
THEN REDUCE_ARITH_TAC
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[];(*4*)
EXPAND_TAC"a3"
THEN EXPAND_TAC"a4"
THEN REWRITE_TAC[VECTOR_ARITH`-- A+B= B-A:real^3`;VECTOR_ARITH`(&1+A-B)%X+B%U-A%V=X-A%(V-X)+B%(U-X)`]
THEN VECTOR_ARITH_TAC]]]]);;














let AFF_GT_CUT_XFAN_IMP_EDGE_FAN=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\  ~(aff_gt{x} {v,w} INTER {v | ?e. e IN E /\ v IN aff_ge {x} e}={})
==> {v,w} IN E`,


REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"EM")
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> MP_TAC (th) THEN ASSUME_TAC(th))
THEN REWRITE_TAC[fan80]
THEN DISCH_THEN(LABEL_TAC"BE")
THEN DISCH_THEN(LABEL_TAC"YEU")
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[SET_RULE`~(A={})<=> ?x'. x' IN A`;INTER;IN_ELIM_THM]
THEN STRIP_TAC 
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC expand_edge_graph_fan [`(x:real^3)`;` (V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(e:real^3->bool)`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"A")
THEN DISCH_TAC
THEN REMOVE_THEN "A" MP_TAC
THEN POP_ASSUM(fun th->REWRITE_TAC[(th)])
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"A")
THEN DISCH_TAC
THEN REMOVE_THEN "A" MP_TAC
THEN REMOVE_THEN "BE" (fun th->  MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC"BE") )
THEN MRESAL_TAC not_collinear_is_properties_fully_surrounded1 [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`&1:real`][REAL_ARITH`&0<= &1/\ &1<= &1 `;VECTOR_ARITH`(&1 - &1) % u + &1 % w=w`]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(w:real^3)`;]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w':real^3)`;
` (v':real^3)`]  
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v':real^3`;`w':real^3`][IN_ELIM_THM]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "VUT1")
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`w:real^3`][IN_ELIM_THM]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "VUT")
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "VUT1")
THEN DISCH_TAC THEN DISCH_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(t3'= &0) \/ ~(t3':real= &0)`)
THENL(*1*)[

ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN REDUCE_VECTOR_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(t2'= &0) \/ ~(t2':real= &0)`)
THENL(*2*)[
ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN REDUCE_VECTOR_TAC
THEN DISCH_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(REAL_ARITH`&0<t3:real==> ~(t3= &0)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(t3:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN FIND_ASSUM (fun th-> REWRITE_TAC[SYM(th)])`t1 + t2 + t3 = &1:real`
THEN REWRITE_TAC[VECTOR_ARITH`t1 % x + t2 % v + t3 % w =
 (t1 + t2 + t3) % x<=>(t3) % (w- x)=(-- t2) % (v-x) `]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
(t3) % (w- x)=(-- t2) % (v-x)
==> (inv (t3))%((t3) % (w- x) ) = (inv (t3))%((-- t2) % (v-x)):real^3`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`;]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[VECTOR_ARITH`A-B=C%(D-B)<=>A=(&1-C)%B+C%D`;]
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC `~(w IN aff {x, v:real^3})`
THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM]
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`&1-(inv t3 * (--t2)):real`
THEN EXISTS_TAC`(inv t3 * (--t2)):real`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC
;(*2*)

FIND_ASSUM MP_TAC`t1 + t2 + t3 = &1:real`
THEN REWRITE_TAC[REAL_ARITH`t1 + t2 + t3 = &1<=> t1= &1 -t2 -t3`; REAL_ARITH`A+B= &1<=> A= &1-B`]
THEN DISCH_THEN(LABEL_TAC"MAI")
THEN DISCH_THEN (LABEL_TAC"YEU EM")
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(&1 - t2 - t3) % x + t2 % v + t3 % w =
 (&1 - t2') % x + t2' % v'<=> t2' % v'=(t2' - t2 - t3) % x + t2 % v + t3 % w:real^3 `]
THEN MP_TAC(ISPEC`(t2':real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2' % v'=(t2' - t2 - t3) % x + t2 % v + t3 % w:real^3
==> (inv (t2'))%(t2' % v' ) = (inv (t2'))%((t2' - t2 - t3) % x + t2 % v + t3 % w:real^3)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C+D)=(A%B+A%C+A%D)`;REAL_ARITH`a*(b-c-d)=a*b-a*c-a*d`]
THEN REDUCE_VECTOR_TAC
THEN DISCH_TAC
THEN SUBGOAL_THEN `~coplanar{x:real^3,v':real^3,v:real^3,u:real^3}` ASSUME_TAC
THENL(*3*)[

POP_ASSUM MP_TAC
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3, v', v, u}= {(--x) +x, (--x) +v', (--x) +v, (--x) +u}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3,v':real^3,v:real^3,u:real^3}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)] THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,C,B}`])
THEN DISCH_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th;VECTOR_ARITH`((&1 - inv t2' * t2 - inv t2' * t3) % x +
                         (inv t2' * t2) % v +
                         (inv t2' * t3) % w) -
                        x=(inv t2' * t2) % (v-x) +
                         (inv t2' * t3) % (w-x)`])
THEN MP_TAC(REAL_ARITH`&0< t2 /\ &0< t3 /\ t1+t2+t3= &1==>  &0 < &1- t1 /\ ~(&1- t1 = &0)/\ t2+t3= &1- t1`)
THEN REMOVE_THEN"MAI" MP_TAC
THEN DISCH_TAC
THEN REMOVE_ASSUM_TAC
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ISPEC`&1- (t1:real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`&1- (t1:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2 + t3 = &1 - t1:real
==> (inv ( &1 - t1))*(t2 + t3) = (inv ( &1 - t1))*( &1 - t1:real)
`)

THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C)= &1<=> A*B = &1 -A*C`]
THEN STRIP_TAC
THEN MP_TAC(REAL_ARITH` ~(t2' = &0) /\ &0 <= t2'==> &0 <t2'`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`t2':real`;`inv (&1- t1):real`]
THEN MRESA_TAC REAL_LT_MUL[`inv (&1- t1):real`;`t3:real`;]
THEN MP_TAC(REAL_ARITH`&0< (inv (&1- t1)) * t3 ==> ~( (inv(&1- t1)) * t3= &0)`)
THEN MP_TAC(REAL_ARITH`&0< (t2') * inv (&1- t1) ==> ~((t2')* inv(&1- t1)= &0)`)
THEN RESA_TAC
THEN ASSUME_TAC(REAL_ARITH`~(&1 = &0)`)
THEN MRESA_TAC COPLANAR_SCALE_ALL [`&1`;`&1`;`(t2')* inv(&1- t1)`;`u - x:real^3`; `v - x:real^3`; `(inv t2' * t2) % (v - x) +(inv t2' * t3) % (w - x):real^3`]
THEN POP_ASSUM (fun th -> ONCE_REWRITE_TAC[SYM(th);])
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%(B+D)=A%B+A%D`;
VECTOR_ARITH`A%B%C=(A*B)%C`; REAL_ARITH`(t2' * inv (&1 - t1)) * inv t2' * t2= (inv (t2')* t2' ) * inv (&1-t1) * t2`]
THEN REDUCE_ARITH_TAC
THEN DISCH_TAC
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v, w}= {(--x) +x, (--x) +v, (--x) +u, (--x) + w}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, w}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)] )
THEN DISCH_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN MRESA_TAC continuous_coplanar_fan[`(vec 0):real^3 `;` u-x:real^3`;` v-x:real^3`;` w-x:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th`(inv(&1- t1)) * t3:real`)
;(*3*)


POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"DICH")
THEN DISCH_TAC
THEN MP_TAC(REAL_ARITH` ~(t2' = &0) /\ &0 <= t2'==> &0 <t2'`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`t2':real`
THEN MRESA_TAC REAL_LT_MUL[`inv (t2'):real`;`t2:real`;]
THEN MRESA_TAC REAL_LT_MUL[`inv (t2'):real`;`t3:real`;]
THEN SUBGOAL_THEN `(v':real^3) IN aff_gt {x} {v,w:real^3}` ASSUME_TAC
THENL(*4*)[

ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC`(&1 - inv t2' * t2 - inv t2' * t3:real)`
THEN EXISTS_TAC`inv t2' * t2:real`
THEN EXISTS_TAC`(inv t2' * t3:real)`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC;(*4*)


REMOVE_THEN "DICH" MP_TAC
THEN REMOVE_THEN "VUT" MP_TAC
THEN DISCH_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_THEN "EM"(fun th-> MRESA1_TAC th `v':real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN MRESA_TAC exists_element_in_half_sapace_fan
[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v':real^3`;`  v:real^3`;`u:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u':real^3)`;
` (v':real^3)`]
THEN MRESAL_TAC exists_cut_small_edges_fan_le[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`&1`;`v':real^3`;`u':real^3`][REAL_ARITH`&0< &1/\ &1<= &1`;VECTOR_ARITH`(&1 - &1) % u + &1 % w=w`]
THEN DISCH_TAC
THEN REMOVE_ASSUM_TAC
THEN MRESAL_TAC not_cut_in_edges_fan [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`&1 -t:real`]
[REAL_ARITH`(&0< &1- t<=> t< &1)/\ (&1- t< &1<=> &0< t)`;VECTOR_ARITH`(&1 - (&1 - t)) % u + (&1 - t) % w=(&1 - t) % w +t % u`;SET_RULE`aff_gt {x} {v, (&1 - t) % w + t % u} INTER
      {v | ?e. e IN E /\ v IN aff_ge {x} e} =
      {}<=> !e. e IN E ==> aff_gt {x} {v, (&1 - t) % w + t % u} INTER
      aff_ge {x} e =
      {}`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{v',u':real^3}`)
THEN MRESA_TAC aff_gt_subset_aff_ge[`x:real^3`;`v':real^3`;`u':real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[]

](*4*)](*3*)](*2*);(*1*)


DISJ_CASES_TAC(REAL_ARITH`(t2'= &0) \/ ~(t2':real= &0)`)

THENL(*2*)[

FIND_ASSUM MP_TAC`t1 + t2 + t3 = &1:real`
THEN REWRITE_TAC[REAL_ARITH`t1 + t2 + t3 = &1<=> t1= &1 -t2 -t3`; REAL_ARITH`A+B= &1<=> A= &1-B`]
THEN DISCH_THEN(LABEL_TAC"MAI")
THEN DISCH_THEN (LABEL_TAC"YEU EM")
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(&1 - t2 - t3) % x + t2 % v + t3 % w =
 (&1 - &0 - t3') % x + &0% v'+t3' % w'<=> t3' % w'=(t3' - t2 - t3) % x + t2 % v + t3 % w:real^3 `]
THEN MP_TAC(ISPEC`(t3':real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t3' % w'=(t3' - t2 - t3) % x + t2 % v + t3 % w:real^3
==> (inv (t3'))%(t3' % w' ) = (inv (t3'))%((t3' - t2 - t3) % x + t2 % v + t3 % w:real^3)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C+D)=(A%B+A%C+A%D)`;REAL_ARITH`a*(b-c-d)=a*b-a*c-a*d`]
THEN REDUCE_VECTOR_TAC
THEN DISCH_TAC
THEN SUBGOAL_THEN `~coplanar{x:real^3,w':real^3,v:real^3,u:real^3}` ASSUME_TAC
THENL(*3*)[

 POP_ASSUM MP_TAC
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3, w', v, u}= {(--x) +x, (--x) +w', (--x) +v, (--x) +u}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3,w':real^3,v:real^3,u:real^3}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)] THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,C,B}`])
THEN DISCH_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th;VECTOR_ARITH`((&1 - inv t2' * t2 - inv t2' * t3) % x +
                         (inv t2' * t2) % v +
                         (inv t2' * t3) % w) -
                        x=(inv t2' * t2) % (v-x) +
                         (inv t2' * t3) % (w-x)`])
THEN MP_TAC(REAL_ARITH`&0< t2 /\ &0< t3 /\ t1+t2+t3= &1==>  &0 < &1- t1 /\ ~(&1- t1 = &0)/\ t2+t3= &1- t1`)
THEN REMOVE_THEN"MAI" MP_TAC
THEN DISCH_TAC
THEN REMOVE_ASSUM_TAC
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ISPEC`&1- (t1:real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`&1- (t1:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2 + t3 = &1 - t1:real
==> (inv ( &1 - t1))*(t2 + t3) = (inv ( &1 - t1))*( &1 - t1:real)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C)= &1<=> A*B = &1 -A*C`]
THEN STRIP_TAC
THEN MP_TAC(REAL_ARITH` ~(t3' = &0) /\ &0 <= t3'==> &0 <t3'`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`t3':real`;`inv (&1- t1):real`]
THEN MRESA_TAC REAL_LT_MUL[`inv (&1- t1):real`;`t3:real`;]
THEN MP_TAC(REAL_ARITH`&0< (inv (&1- t1)) * t3 ==> ~( (inv(&1- t1)) * t3= &0)`)
THEN MP_TAC(REAL_ARITH`&0< (t3') * inv (&1- t1) ==> ~((t3')* inv(&1- t1)= &0)`)
THEN RESA_TAC
THEN ASSUME_TAC(REAL_ARITH`~(&1 = &0)`)
THEN MRESA_TAC COPLANAR_SCALE_ALL [`&1`;`&1`;`(t3')* inv(&1- t1)`;`u - x:real^3`; `v - x:real^3`; `(inv t3' * t2) % (v - x) +(inv t3' * t3) % (w - x):real^3`]
THEN POP_ASSUM (fun th -> ONCE_REWRITE_TAC[SYM(th);])
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%(B+D)=A%B+A%D`;
VECTOR_ARITH`A%B%C=(A*B)%C`; REAL_ARITH`(t2' * inv (&1 - t1)) * inv t2' * t2= (inv (t2')* t2' ) * inv (&1-t1) * t2`]
THEN REDUCE_ARITH_TAC
THEN DISCH_TAC
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v, w}= {(--x) +x, (--x) +v, (--x) +u, (--x) +w}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, w}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN RESA_TAC
THEN MRESA_TAC continuous_coplanar_fan[`(vec 0):real^3 `;` u-x:real^3`;` v-x:real^3`;` w-x:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN RESA_TAC
THEN POP_ASSUM (fun th-> MRESA1_TAC th`(inv(&1- t1)) * t3:real`)

;(*3*)


POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"DICH")
THEN DISCH_TAC
THEN MP_TAC(REAL_ARITH` ~(t3' = &0) /\ &0 <= t3'==> &0 <t3'`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`t3':real`
THEN MRESA_TAC REAL_LT_MUL[`inv (t3'):real`;`t2:real`;]
THEN MRESA_TAC REAL_LT_MUL[`inv (t3'):real`;`t3:real`;]
THEN SUBGOAL_THEN `(w':real^3) IN aff_gt {x} {v,w:real^3}` ASSUME_TAC
THENL(*4*)[

ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC`(&1 - inv t3' * t2 - inv t3' * t3:real)`
THEN EXISTS_TAC`inv t3' * t2:real`
THEN EXISTS_TAC`(inv t3' * t3:real)`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC;(*4*)

REMOVE_THEN "DICH" MP_TAC
THEN REMOVE_THEN "VUT" MP_TAC
THEN DISCH_TAC
THEN REMOVE_ASSUM_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (v':real^3)`;
` (w':real^3)`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN RESA_TAC
THEN REMOVE_THEN "EM"(fun th-> MRESA1_TAC th `w':real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN MRESA_TAC exists_element_in_half_sapace_fan
[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`w':real^3`;`  v:real^3`;`u:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u':real^3)`;
` (w':real^3)`]
THEN MRESAL_TAC exists_cut_small_edges_fan_le[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`&1`;`w':real^3`;`u':real^3`][REAL_ARITH`&0< &1/\ &1<= &1`;VECTOR_ARITH`(&1 - &1) % u + &1 % w=w`]
THEN DISCH_TAC
THEN REMOVE_ASSUM_TAC
THEN MRESAL_TAC not_cut_in_edges_fan [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`&1 -t:real`]
[REAL_ARITH`(&0< &1- t<=> t< &1)/\ (&1- t< &1<=> &0< t)`;VECTOR_ARITH`(&1 - (&1 - t)) % u + (&1 - t) % w=(&1 - t) % w +t % u`;SET_RULE`aff_gt {x} {v, (&1 - t) % w + t % u} INTER
      {v | ?e. e IN E /\ v IN aff_ge {x} e} =
      {}<=> !e. e IN E ==> aff_gt {x} {v, (&1 - t) % w + t % u} INTER
      aff_ge {x} e =
      {}`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{w',u':real^3}`)
THEN MRESA_TAC aff_gt_subset_aff_ge[`x:real^3`;`w':real^3`;`u':real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[]](*4*)](*3*);(*2*)






ASM_REWRITE_TAC[]
THEN DISCH_TAC THEN DISCH_THEN(LABEL_TAC"BO")
THEN MP_TAC(REAL_ARITH` ~(t3' = &0) /\ &0 <= t3'==> &0 <t3'`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH` ~(t2' = &0) /\ &0 <= t2'==> &0 <t2'`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`t2':real`
THEN MRESA1_TAC REAL_MUL_LINV`t2':real`
THEN MRESA_TAC REAL_LT_MUL[`inv (t2'):real`;`t2:real`;]
THEN MRESA_TAC REAL_LT_MUL[`inv (t2'):real`;`t3:real`;]
THEN MRESA1_TAC REAL_LT_INV`t3':real`
THEN MRESA1_TAC REAL_MUL_LINV`t3':real`
THEN MRESA_TAC REAL_LT_MUL[`inv (t3'):real`;`t2:real`;]
THEN MRESA_TAC REAL_LT_MUL[`inv (t3'):real`;`t3:real`;]
THEN SUBGOAL_THEN`(x':real^3) IN aff_gt {x} {v',w'}` ASSUME_TAC
THENL(*3*)[
MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v':real^3`;`w':real^3`][IN_ELIM_THM]
THEN ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC`t1':real`
THEN EXISTS_TAC`t2':real`
THEN EXISTS_TAC`t3':real`
THEN ASM_REWRITE_TAC[]
;(*3*)


SUBGOAL_THEN`(x':real^3) IN aff_ge {x} {v',w'}` ASSUME_TAC
THENL(*4*)[
ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC`t1':real`
THEN EXISTS_TAC`t2':real`
THEN EXISTS_TAC`t3':real`
THEN ASM_REWRITE_TAC[];(*4*)

SUBGOAL_THEN `(x':real^3) IN aff_gt {x} {v,w:real^3}` ASSUME_TAC

THENL(*5*)[
ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3:real`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC;(*5*)

SUBGOAL_THEN `(x':real^3) IN aff_ge {x} {v,w:real^3}` ASSUME_TAC
THENL(*6*)[
MRESA_TAC AFF_GE_1_2[`x:real^3`;`v:real^3`;`w:real^3`]
THEN ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3:real`
THEN MP_TAC(REAL_ARITH`&0< t2==> &0 <= t2`) THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0< t3==> &0 <= t3`) THEN RESA_TAC
THEN ASM_REWRITE_TAC[];(*6*)


FIND_ASSUM MP_TAC`{v',w':real^3} IN E`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN DISCH_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (v':real^3)`;
` (w':real^3)`]  
THEN SUBGOAL_THEN `~collinear {x, x', w':real^3}` ASSUME_TAC
THENL(*7*)[

ASM_REWRITE_TAC[collinear1_fan;]
THEN FIND_ASSUM MP_TAC `~((v':real^3) IN aff{x,w'})`
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff; AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % v' + t3' % w' = u' % x + v'' % w'
<=> t2' % v' = (u'-t1') % x + (v''-t3') % w'`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2' % v' = (u'-t1') % x + (v''-t3') % w':real^3
==> (inv ( t2'))%(t2' % v') = (inv (t2'))%((u'-t1') % x + (v''-t3') % w')
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C= (A*B)%C`;VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC `(inv t2' * (u' - t1':real))`
THEN EXISTS_TAC `(inv t2' * (v'' - t3':real))`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv t2' * (u' - t1') + inv t2' * (v'' - t3')=inv t2'* (t2'+ (u'+v'') - (t1'+t2'+t3'))`;REAL_ARITH`t2'+ &1 - &1=t2'`]
;(*7*)



SUBGOAL_THEN `~collinear {x, x', v':real^3}` ASSUME_TAC

THENL(*8*)[
ASM_REWRITE_TAC[collinear1_fan;]
THEN FIND_ASSUM MP_TAC `~((w':real^3) IN aff{x,v'})`
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff; AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % v' + t3' % w' = u' % x + v'' % v'
<=> t3' % w' = (u'-t1') % x + (v''-t2') % v'`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t3' % w' = (u'-t1') % x + (v''-t2') % v':real^3
==> (inv ( t3'))%(t3' % w') = (inv (t3'))%((u'-t1') % x + (v''-t2') % v')
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C= (A*B)%C`;VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC `(inv t3' * (u' - t1':real))`
THEN EXISTS_TAC `(inv t3' * (v'' - t2':real))`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv t3' * (u' - t1') + inv t3' * (v'' - t2')=inv t3'* (t3'+ (u'+v'') - (t1'+t2'+t3'))`;REAL_ARITH`t3'+ &1 - &1=t3'`]
;(*8*)



SUBGOAL_THEN`~collinear {x, x', w:real^3}` ASSUME_TAC

THENL(*9*)[
REMOVE_THEN "BO" MP_TAC
THEN DISCH_TAC THEN REMOVE_ASSUM_TAC
THEN ASM_REWRITE_TAC[collinear1_fan;]
THEN FIND_ASSUM MP_TAC`~(w IN aff {x, v:real^3})`
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1 % x + t2 % v + t3 % (va) =
      u' % x + v' % (va)<=>
(&1-(t1+t2+t3)) % x  + (v'-t3) % (va-x) =
      (&1-(u'+v') ) % x + t2 % (v-x)`;REAL_ARITH`a-a= &0`]
THEN REDUCE_VECTOR_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(v'' - t3 = &0) \/ ~(v'' - (t3:real)= &0)`)
THENL(*10*)[
ASM_REWRITE_TAC[VECTOR_ARITH`&0 %A=B<=> B= vec 0`;VECTOR_MUL_EQ_0]
THEN MP_TAC(REAL_ARITH`&0< t2==> ~(t2 = &0)`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A-B= vec 0<=> A=B`];(*10*)

MRESA1_TAC REAL_MUL_LINV `(v''-t3:real)`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(v'' - t3) % (w- x) = t2 % (v - x:real^3) ==> (inv (v'' -t3)) % ((v'' - t3) % (w - x)) = (inv (v'' -t3)) % ( t2 % (v - x))`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th]
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(A%B%C=(A*B)%C:real^3)`])
THEN REWRITE_TAC[VECTOR_ARITH`&1 % (va - x) = (inv (v' - t3) * t2) % (v - x)<=>
va  = (&1-(inv (v' - t3) * t2))%x +(inv (v' - t3) * t2) % v:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`&1-(inv (v'' - t3) * (t2:real))`
THEN EXISTS_TAC`(inv (v'' - t3) * (t2:real))`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC](*10*);(*9*)

SUBGOAL_THEN `~collinear {x, x', v:real^3}` ASSUME_TAC
THENL(*10*)[
REMOVE_THEN "BO" MP_TAC
THEN DISCH_TAC
THEN REMOVE_ASSUM_TAC
THEN ASM_REWRITE_TAC[collinear1_fan]
THEN FIND_ASSUM MP_TAC `~(w IN aff {x, v:real^3})`
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff; AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % v' + t3' % w' = u' % x + v'' % v'
<=> t3' % w' = (u'-t1') % x + (v''-t2') % v'`]
THEN MP_TAC(REAL_ARITH`&0< t3 ==> ~(t3= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`t3:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t3 % w = (u'-t1) % x + (v''-t2) % v:real^3
==> (inv ( t3))%(t3 % w) = (inv (t3))%((u'-t1) % x + (v''-t2) % v)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C= (A*B)%C`;VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC `(inv t3 * (u' - t1:real))`
THEN EXISTS_TAC `(inv t3 * (v'' - t2:real))`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv t3' * (u' - t1') + inv t3' * (v'' - t2')=inv t3'* (t3'+ (u'+v'') - (t1'+t2'+t3'))`;REAL_ARITH`t3'+ &1 - &1=t3'`]
;(*10*)

REMOVE_THEN "VUT1" MP_TAC
THEN REMOVE_THEN "VUT1" MP_TAC
THEN REMOVE_THEN "BO" MP_TAC
THEN REMOVE_THEN "VUT" MP_TAC
THEN DISCH_TAC THEN DISCH_TAC THEN DISCH_TAC THEN DISCH_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN DISJ_CASES_TAC(SET_RULE`(&0 < azim x (x':real^3) v w' /\
      azim x (x') v w' < pi)\/ ~(&0 < azim x (x') v w' /\
      azim x (x') v w' < pi)`)
THENL(*11*)[
MRESAL_TAC exists_cut_small_edges_fan_le[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`&1:real`;`x':real^3`;`w':real^3`][REAL_ARITH`&0< &1/\ &1<= &1`;VECTOR_ARITH`(&1 - &1) % u + &1 % w=w`]
THEN MRESAL_TAC not_cut_in_edges_fan [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`&1 -t:real`]
[REAL_ARITH`(&0< &1- t<=> t< &1)/\ (&1- t< &1<=> &0< t)`;VECTOR_ARITH`(&1 - (&1 - t)) % u + (&1 - t) % w=(&1 - t) % w +t % u`;SET_RULE`aff_gt {x} {v, (&1 - t) % w + t % u} INTER
      {v | ?e. e IN E /\ v IN aff_ge {x} e} =
      {}<=> !e. e IN E ==> aff_gt {x} {v, (&1 - t) % w + t % u} INTER
      aff_ge {x} e =
      {}`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{v',w':real^3}`)
THEN MRESA_TAC aff_gt1_subset_aff_ge[`x:real^3`;`v':real^3`;`w':real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[]
;(*11*)

ASSUME_TAC(ISPECL[`x:real^3`;`v':real^3`;`w':real^3`;`x':real^3`]azim)
THEN ASSUME_TAC(PI_WORKS)
THEN MP_TAC(REAL_ARITH`~(&0 < azim x x' v w' /\ azim x x' v w' < pi)/\
&0 <= azim x x' v w' /\ &0< pi==>   pi< azim x x' v w' \/(azim x x' v w'= &0 \/  azim x x' v w'=pi) `)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*12*)(*3GOAL*)[

POP_ASSUM MP_TAC
THEN MRESA_TAC aff_gt2_subset_aff_ge[`x:real^3`;`v':real^3`;`w':real^3 `;`x':real^3`]
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN DISCH_TAC
THEN MP_TAC(REAL_ARITH`azim x x' v' w' < azim x x' v w'==>azim x x' v' w' <= azim x x' v w'`)
THEN RESA_TAC
THEN MRESA_TAC sum5_azim_fan[`x:real^3`;`x':real^3`;`v:real^3`;`v':real^3`;`w':real^3`]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC aff_gt2_subset_aff_ge[`x:real^3`;`v':real^3`;`w':real^3 `;`x':real^3`]
THEN DISCH_TAC THEN DISCH_TAC
THEN MP_TAC(REAL_ARITH`pi < azim x x' v w' /\ azim x x' v w' = azim x x' v v' + pi
/\  azim x x' v w' < &2 * pi==> &0 <azim x x' v v' /\ azim x x' v v' < pi`)
THEN RESA_TAC
THEN MRESAL_TAC exists_cut_small_edges_fan_le[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`&1:real`;`x':real^3`;`v':real^3`][REAL_ARITH`&0< &1/\ &1<= &1`;VECTOR_ARITH`(&1 - &1) % u + &1 % w=w`]
THEN MRESAL_TAC not_cut_in_edges_fan [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`&1 -t:real`]
[REAL_ARITH`(&0< &1- t<=> t< &1)/\ (&1- t< &1<=> &0< t)`;VECTOR_ARITH`(&1 - (&1 - t)) % u + (&1 - t) % w=(&1 - t) % w +t % u`;SET_RULE`aff_gt {x} {v, (&1 - t) % w + t % u} INTER
      {v | ?e. e IN E /\ v IN aff_ge {x} e} =
      {}<=> !e. e IN E ==> aff_gt {x} {v, (&1 - t) % w + t % u} INTER
      aff_ge {x} e =
      {}`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{w',v':real^3}`)
THEN MRESA_TAC aff_gt1_subset_aff_ge[`x:real^3`;`w':real^3`;`v':real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];(*12*)

MRESA_TAC AZIM_EQ_0[`x:real^3`;`x':real^3`;`v:real^3`;`w':real^3`]
THEN MRESA_TAC AZIM_EQ_0_ALT[`x:real^3`;`x':real^3`;`v:real^3`;`w':real^3`]
THEN MRESA_TAC AZIM_EQ_0[`x:real^3`;`x':real^3`;`w':real^3`;`v:real^3`]
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`x':real^3`;`w':real^3`;`v:real^3`]
THEN DISJ_CASES_TAC(SET_RULE`w' IN aff_gt {x} {v,w:real^3}\/ ~(w' IN aff_gt {x} {v,w:real^3})`)
THENL(*13*)[

MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`w:real^3`][IN_ELIM_THM]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"DICH CHUYEN")
THEN DISCH_TAC
THEN REMOVE_THEN "DICH CHUYEN"(fun th->MP_TAC(th) THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN REMOVE_ASSUM_TAC
 THEN ASSUME_TAC(th))
THEN STRIP_TAC 
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`w' = t1'' % x + t2'' % v + t3'' % va<=>
 ((t1''+t2''+t3'')- &1) % x + t2'' % (v-x) + t3'' % (va-x)= w' -x:real^3`;REAL_ARITH`&1- &1 = &0`]
THEN REDUCE_VECTOR_TAC
THEN MP_TAC(REAL_ARITH`&0< t2'' /\ &0< t3'' /\ t1''+t2''+t3''= &1==>  &0 < &1- t1'' /\ ~(&1- t1'' = &0)/\ t2''+t3''= &1- t1''`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ISPEC`&1- (t1'':real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`&1- (t1'':real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2'' + t3'' = &1 - t1'':real
==> (inv ( &1 - t1''))*(t2'' + t3'') = (inv ( &1 - t1''))*( &1 - t1'':real)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C)= &1<=> &1- A*C = A*B`]
THEN STRIP_TAC
THEN MRESA_TAC REAL_LT_MUL[`inv(&1- t1'':real)`;`t3'':real`]
THEN MP_TAC(REAL_ARITH`&0<inv(&1- t1'':real)*(t3'':real)==> ~(inv(&1- t1'':real)*(t3'':real)= &0)`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0<inv(&1- t1'':real)==> ~(inv(&1- t1'':real)= &0)`)
THEN RESA_TAC
THEN ASSUME_TAC(REAL_ARITH`~(&1= &0)`)
THEN STRIP_TAC
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v, w}= {(--x) +x, (--x) +v, (--x) +u, (--x) + w}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, w}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN ASM_REWRITE_TAC[]
THEN DISCH_TAC
THEN MRESA_TAC continuous_coplanar_fan[`(vec 0):real^3 `;` u-x:real^3`;` v-x:real^3`;` w-x:real^3`]
THEN POP_ASSUM (fun th-> MRESAL1_TAC th`(inv(&1- t1'')) * t3'':real`[VECTOR_ARITH`(A*B)%X+(A*C)%Y=A%(B%X+C%Y)`])
THEN MRESAL_TAC COPLANAR_SCALE_ALL [`&1`;`&1`;`inv(&1- t1'')`;`u - x:real^3`; `v - x:real^3`; `w'-x:real^3`]
[VECTOR_ARITH`&1 %A=A`]
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v,w'}= {(--x) +x, (--x) +u, (--x) +v, (--x) + w'}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, w'}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,B,C}`]
THEN DISCH_TAC
THEN REMOVE_THEN "EM"(fun th-> MRESA1_TAC th `w':real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN MRESA_TAC exists_element_in_half_sapace_fan
[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`w':real^3`;`  v:real^3`;`u:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u':real^3)`;
` (w':real^3)`]
THEN MRESAL_TAC exists_cut_small_edges_fan_le[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`&1:real`;`w':real^3`;`u':real^3`][REAL_ARITH`&0< &1/\ &1<= &1`;VECTOR_ARITH`(&1 - &1) % u + &1 % w=w`]
THEN MRESAL_TAC not_cut_in_edges_fan [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`&1 -t:real`]
[REAL_ARITH`(&0< &1- t<=> t< &1)/\ (&1- t< &1<=> &0< t)`;VECTOR_ARITH`(&1 - (&1 - t)) % u + (&1 - t) % w=(&1 - t) % w +t % u`;SET_RULE`aff_gt {x} {v, (&1 - t) % w + t % u} INTER
      {v | ?e. e IN E /\ v IN aff_ge {x} e} =
      {}<=> !e. e IN E ==> aff_gt {x} {v, (&1 - t) % w + t % u} INTER
      aff_ge {x} e =
      {}`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{w',u':real^3}`)
THEN MRESA_TAC aff_gt_subset_aff_ge[`x:real^3`;`w':real^3`;`u':real^3`;]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];(*13*)


DISJ_CASES_TAC(SET_RULE`v' IN aff_gt {x} {v,w:real^3}\/ ~(v' IN aff_gt {x} {v,w:real^3})`)
THENL(*14*)[

MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`w:real^3`][IN_ELIM_THM]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"DICH CHUYEN")
THEN DISCH_TAC
THEN REMOVE_THEN "DICH CHUYEN"(fun th->MP_TAC(th) THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN REMOVE_ASSUM_TAC
 THEN ASSUME_TAC(th))
THEN STRIP_TAC 
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`v' = t1'' % x + t2'' % v + t3'' % va<=>
 ((t1''+t2''+t3'')- &1) % x + t2'' % (v-x) + t3'' % (va-x)= v' -x:real^3`;REAL_ARITH`&1- &1 = &0`]
THEN REDUCE_VECTOR_TAC
THEN MP_TAC(REAL_ARITH`&0< t2'' /\ &0< t3'' /\ t1''+t2''+t3''= &1==>  &0 < &1- t1'' /\ ~(&1- t1'' = &0)/\ t2''+t3''= &1- t1''`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ISPEC`&1- (t1'':real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`&1- (t1'':real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2'' + t3'' = &1 - t1'':real
==> (inv ( &1 - t1''))*(t2'' + t3'') = (inv ( &1 - t1''))*( &1 - t1'':real)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C)= &1<=> &1- A*C = A*B`]
THEN STRIP_TAC
THEN MRESA_TAC REAL_LT_MUL[`inv(&1- t1'':real)`;`t3'':real`]
THEN MP_TAC(REAL_ARITH`&0<inv(&1- t1'':real)*(t3'':real)==> ~(inv(&1- t1'':real)*(t3'':real)= &0)`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0<inv(&1- t1'':real)==> ~(inv(&1- t1'':real)= &0)`)
THEN RESA_TAC
THEN ASSUME_TAC(REAL_ARITH`~(&1= &0)`)
THEN STRIP_TAC
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN STRIP_TAC
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v, w}= {(--x) +x, (--x) +v, (--x) +u, (--x) +w}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, w}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN STRIP_TAC
THEN MRESA_TAC continuous_coplanar_fan[`(vec 0):real^3 `;` u-x:real^3`;` v-x:real^3`;` w-x:real^3`]
THEN POP_ASSUM (fun th-> MRESAL1_TAC th`(inv(&1- t1'')) * t3'':real`[VECTOR_ARITH`(A*B)%X+(A*C)%Y=A%(B%X+C%Y)`])
THEN MRESAL_TAC COPLANAR_SCALE_ALL [`&1`;`&1`;`inv(&1- t1'')`;`u - x:real^3`; `v - x:real^3`; `v'-x:real^3`]
[VECTOR_ARITH`&1 %A=A`]
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v,v'}= {(--x) +x, (--x) +u, (--x) +v, (--x) + v'}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, v'}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,B,C}`]
THEN DISCH_TAC
THEN REMOVE_THEN "EM"(fun th-> MRESA1_TAC th `v':real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN MRESA_TAC exists_element_in_half_sapace_fan
[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v':real^3`;`  v:real^3`;`u:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u':real^3)`;
` (v':real^3)`]
THEN MRESAL_TAC exists_cut_small_edges_fan_le[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`&1:real`;`v':real^3`;`u':real^3`][REAL_ARITH`&0< &1/\ &1<= &1`;VECTOR_ARITH`(&1 - &1) % u + &1 % w=w`]
THEN MRESAL_TAC not_cut_in_edges_fan [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`&1 -t:real`]
[REAL_ARITH`(&0< &1- t<=> t< &1)/\ (&1- t< &1<=> &0< t)`;VECTOR_ARITH`(&1 - (&1 - t)) % u + (&1 - t) % w=(&1 - t) % w +t % u`;SET_RULE`aff_gt {x} {v, (&1 - t) % w + t % u} INTER
      {v | ?e. e IN E /\ v IN aff_ge {x} e} =
      {}<=> !e. e IN E ==> aff_gt {x} {v, (&1 - t) % w + t % u} INTER
      aff_ge {x} e =
      {}`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{v',u':real^3}`)
THEN MRESA_TAC aff_gt_subset_aff_ge[`x:real^3`;`v':real^3`;`u':real^3`;]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];(*14*)


SUBGOAL_THEN`~(w' IN aff_gt {x} { x',v:real^3})` ASSUME_TAC

THENL(*15*)[
REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN MRESA_TAC aff_gt3_subset_aff_gt[`x:real^3`;`v:real^3`;`w:real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
THEN SET_TAC[]
;(*15*)

MP_TAC(ISPECL[`x:real^3`;`x':real^3`;`v:real^3`;`w':real^3`] decomposition_planar_by_angle_fan)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*16*)[
MRESA_TAC aff_gt1_subset_aff_ge[`x:real^3`;`v':real^3`;`w':real^3`;`x':real^3`]
THEN MP_TAC(SET_RULE`v IN aff_gt {x} {x', w':real^3} /\ aff_gt {x} {x', w'} SUBSET aff_ge {x} {v', w'}
==>v IN aff_ge {x} {v', w':real^3}`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`; `v:real^3`; `u:real^3`; `v':real^3`;`w':real^3`]properties_of_fan7)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*17*)[

MRESA_TAC aff_gt2_subset_aff_ge[`x:real^3`;`v':real^3`; `w':real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th-> ASM_REWRITE_TAC[SYM(th)])
THEN MP_TAC(PI_WORKS)
THEN REAL_ARITH_TAC;(*17*)

MRESA_TAC aff_gt2_subset_aff_ge[`x:real^3`;`v':real^3`; `w':real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th-> ASM_REWRITE_TAC[SYM(th)] 
THEN MRESA_TAC aff_gt2_subset_aff_ge[`x:real^3`;`v:real^3`; `w:real^3`;`x':real^3`]
THEN DISCH_TAC
THEN REWRITE_TAC[th])
THEN MRESA_TAC AZIM_EQ_PI_SYM [`x:real^3`;`x':real^3`; `v':real^3`;`v:real^3`]
THEN ASSUME_TAC(REAL_ARITH`pi <= pi`)
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`x':real^3`;`v:real^3`;`w:real^3`;`v':real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`A=A+B <=> B= &0`]
THEN DISCH_TAC
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`x':real^3`;`w:real^3`;`v':real^3`]
THEN MP_TAC(ISPECL[`x:real^3`;`x':real^3`;`v':real^3`;`w:real^3`] decomposition_planar_by_angle_fan)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*18*)[

MRESA_TAC aff_gt1_subset_aff_gt[`x:real^3`;`v:real^3`; `w:real^3`;`x':real^3`]
THEN ASM_TAC THEN SET_TAC[];(*18*)

MRESA_TAC aff_ge1_subset_aff_ge[`x:real^3`;`w':real^3`; `v':real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN STRIP_TAC THEN STRIP_TAC
THEN MP_TAC(SET_RULE`w IN aff_ge {x} {v', x':real^3} /\ aff_ge {x} {v', x'} SUBSET aff_ge {x} {v', w'}
==>w IN aff_ge {x} {v', w':real^3}`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`; `w:real^3`; `u:real^3`; `v':real^3`;`w':real^3`]properties_of_fan7)
THEN ASM_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN STRIP_TAC
THENL(*19*)[ ASM_REWRITE_TAC[];(*19*)
POP_ASSUM MP_TAC 
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC 
THEN DISCH_THEN(LABEL_TAC "XONG")
THEN DISCH_TAC
THEN REMOVE_THEN "XONG" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC PI_WORKS
THEN REAL_ARITH_TAC
](*19*)](*18*)](*17*);(*16*)

POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN DISCH_TAC
THEN MRESA_TAC aff_ge_subset_aff_gt_union_aff_ge[`(x:real^3)`;`(v:real^3)`;`(x':real^3)`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[]
THEN DISCH_TAC
THEN MP_TAC(SET_RULE`aff_ge {x} {v, x':real^3} SUBSET aff_gt {x, v} {x'} UNION aff_ge {x} {v} /\ w' IN aff_ge {x} {v, x'} ==> w' IN aff_gt {x, v} {x'} UNION aff_ge {x} {v}`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC aff_gt_inter_aff_gt[`(x:real^3)`;`(x':real^3)`;`(v:real^3)`]
THEN ASM_REWRITE_TAC[INTER;IN_ELIM_THM;UNION]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MP_TAC(ISPECL[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`; `w':real^3`; `v':real^3`; `v:real^3`;`u:real^3`]properties1_of_fan7)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> MP_TAC(SYM(th)))
THEN DISCH_TAC
THEN MRESA_TAC point_in_aff_ge[`(x:real^3)`;`(v':real^3)`;`(w':real^3)`]
THEN SUBGOAL_THEN`v IN aff_ge {x} {v', w':real^3}`ASSUME_TAC
THENL(*17*)[
POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[];(*17*)

POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"NHAT")
THEN DISCH_TAC
THEN REMOVE_THEN "NHAT" MP_TAC
THEN DISCH_TAC
THEN MRESA_TAC aff_gt2_subset_aff_ge[`x:real^3`;`v':real^3`; `w':real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th-> ASM_REWRITE_TAC[SYM(th)]
THEN MRESA_TAC aff_gt2_subset_aff_ge[`x:real^3`;`v:real^3`; `w:real^3`;`x':real^3`]
THEN STRIP_TAC THEN REWRITE_TAC[th])
THEN MRESA_TAC AZIM_EQ_PI_SYM [`x:real^3`;`x':real^3`; `v':real^3`;`v:real^3`]
THEN ASSUME_TAC(REAL_ARITH`pi <= pi`)
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`x':real^3`;`v:real^3`;`w:real^3`;`v':real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`A=A+B <=> B= &0`]
THEN DISCH_TAC
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`x':real^3`;`w:real^3`;`v':real^3`]
THEN MP_TAC(ISPECL[`x:real^3`;`x':real^3`;`v':real^3`;`w:real^3`] decomposition_planar_by_angle_fan)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*18*)[

MRESA_TAC aff_gt1_subset_aff_gt[`x:real^3`;`v:real^3`; `w:real^3`;`x':real^3`]
THEN ASM_TAC THEN SET_TAC[];(*18*)

MRESA_TAC aff_ge1_subset_aff_ge[`x:real^3`;`w':real^3`; `v':real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN STRIP_TAC THEN STRIP_TAC
THEN MP_TAC(SET_RULE`w IN aff_ge {x} {v', x':real^3} /\ aff_ge {x} {v', x'} SUBSET aff_ge {x} {v', w'}
==>w IN aff_ge {x} {v', w':real^3}`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`; `w:real^3`; `u:real^3`; `v':real^3`;`w':real^3`]properties_of_fan7)
THEN ASM_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN STRIP_TAC
THENL(*19*)[
ASM_REWRITE_TAC[];(*19*)
POP_ASSUM MP_TAC 
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC 
THEN DISCH_THEN(LABEL_TAC "XONG")
THEN DISCH_TAC
THEN REMOVE_THEN "XONG" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC PI_WORKS
THEN REAL_ARITH_TAC](*19*)](*18*)](*17*)]]]];(*12*)


POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"CUOI")
THEN MRESA_TAC aff_gt2_subset_aff_ge[`x:real^3`;`v':real^3`;`w':real^3 `;`x':real^3`]
THEN ASSUME_TAC(REAL_ARITH`pi<= pi`)
THEN MRESA_TAC sum5_azim_fan[`x:real^3`;`x':real^3`;`v:real^3`;`v':real^3`;`w':real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`A=B+A<=>B= &0`]
THEN DISCH_TAC
THEN MRESA_TAC AZIM_EQ_0[`x:real^3`;`x':real^3`;`v:real^3`;`v':real^3`]
THEN MRESA_TAC AZIM_EQ_0_ALT[`x:real^3`;`x':real^3`;`v:real^3`;`v':real^3`]
THEN MRESA_TAC AZIM_EQ_0[`x:real^3`;`x':real^3`;`v':real^3`;`v:real^3`]
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`x':real^3`;`v':real^3`;`v:real^3`]
THEN DISJ_CASES_TAC(SET_RULE`w' IN aff_gt {x} {v,w:real^3}\/ ~(w' IN aff_gt {x} {v,w:real^3})`)
THENL(*13*)[

MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`w:real^3`][IN_ELIM_THM]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"DICH CHUYEN")
THEN DISCH_TAC
THEN REMOVE_THEN "DICH CHUYEN"(fun th->MP_TAC(th) THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN REMOVE_ASSUM_TAC
 THEN ASSUME_TAC(th))
THEN STRIP_TAC 
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`w' = t1'' % x + t2'' % v + t3'' % va<=>
 ((t1''+t2''+t3'')- &1) % x + t2'' % (v-x) + t3'' % (va-x)= w' -x:real^3`;REAL_ARITH`&1- &1 = &0`]
THEN REDUCE_VECTOR_TAC
THEN MP_TAC(REAL_ARITH`&0< t2'' /\ &0< t3'' /\ t1''+t2''+t3''= &1==>  &0 < &1- t1'' /\ ~(&1- t1'' = &0)/\ t2''+t3''= &1- t1''`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ISPEC`&1- (t1'':real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`&1- (t1'':real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2'' + t3'' = &1 - t1'':real
==> (inv ( &1 - t1''))*(t2'' + t3'') = (inv ( &1 - t1''))*( &1 - t1'':real)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C)= &1<=> &1- A*C = A*B`]
THEN STRIP_TAC
THEN MRESA_TAC REAL_LT_MUL[`inv(&1- t1'':real)`;`t3'':real`]
THEN MP_TAC(REAL_ARITH`&0<inv(&1- t1'':real)*(t3'':real)==> ~(inv(&1- t1'':real)*(t3'':real)= &0)`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0<inv(&1- t1'':real)==> ~(inv(&1- t1'':real)= &0)`)
THEN RESA_TAC
THEN ASSUME_TAC(REAL_ARITH`~(&1= &0)`)
THEN STRIP_TAC
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN DISCH_TAC
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v, w}= {(--x) +x, (--x) +v, (--x) +u, (--x) + w}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, w}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN DISCH_TAC
THEN MRESA_TAC continuous_coplanar_fan[`(vec 0):real^3 `;` u-x:real^3`;` v-x:real^3`;`w-x:real^3`]
THEN POP_ASSUM (fun th-> MRESAL1_TAC th`(inv(&1- t1'')) * t3'':real`[VECTOR_ARITH`(A*B)%X+(A*C)%Y=A%(B%X+C%Y)`])
THEN MRESAL_TAC COPLANAR_SCALE_ALL [`&1`;`&1`;`inv(&1- t1'')`;`u - x:real^3`; `v - x:real^3`; `w'-x:real^3`]
[VECTOR_ARITH`&1 %A=A`]
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v,w'}= {(--x) +x, (--x) +u, (--x) +v, (--x) + w'}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, w'}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,B,C}`]
THEN DISCH_TAC
THEN REMOVE_THEN "EM"(fun th-> MRESA1_TAC th `w':real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN MRESA_TAC exists_element_in_half_sapace_fan
[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`w':real^3`;`  v:real^3`;`u:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u':real^3)`;
` (w':real^3)`]
THEN MRESAL_TAC exists_cut_small_edges_fan_le[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`&1:real`;`w':real^3`;`u':real^3`]
[REAL_ARITH`&0< &1/\ &1<= &1`;VECTOR_ARITH`(&1 - &1) % u + &1 % w=w`]
THEN MRESAL_TAC not_cut_in_edges_fan 
[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`&1 -t:real`]
[REAL_ARITH`(&0< &1- t<=> t< &1)/\ (&1- t< &1<=> &0< t)`;VECTOR_ARITH`(&1 - (&1 - t)) % u + (&1 - t) % w=(&1 - t) % w +t % u`;SET_RULE`aff_gt {x} {v, (&1 - t) % w + t % u} INTER
      {v | ?e. e IN E /\ v IN aff_ge {x} e} =
      {}<=> !e. e IN E ==> aff_gt {x} {v, (&1 - t) % w + t % u} INTER
      aff_ge {x} e =
      {}`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{w',u':real^3}`)
THEN MRESA_TAC aff_gt_subset_aff_ge[`x:real^3`;`w':real^3`;`u':real^3`;]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];(*13*)


DISJ_CASES_TAC(SET_RULE`v' IN aff_gt {x} {v,w:real^3}\/ ~(v' IN aff_gt {x} {v,w:real^3})`)
THENL(*14*)[

MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`w:real^3`][IN_ELIM_THM]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"DICH CHUYEN")
THEN DISCH_TAC
THEN REMOVE_THEN "DICH CHUYEN"(fun th->MP_TAC(th) THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN REMOVE_ASSUM_TAC
 THEN ASSUME_TAC(th))
THEN STRIP_TAC 
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`v' = t1'' % x + t2'' % v + t3'' % va<=>
 ((t1''+t2''+t3'')- &1) % x + t2'' % (v-x) + t3'' % (va-x)= v' -x:real^3`;REAL_ARITH`&1- &1 = &0`]
THEN REDUCE_VECTOR_TAC
THEN MP_TAC(REAL_ARITH`&0< t2'' /\ &0< t3'' /\ t1''+t2''+t3''= &1==>  &0 < &1- t1'' /\ ~(&1- t1'' = &0)/\ t2''+t3''= &1- t1''`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ISPEC`&1- (t1'':real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`&1- (t1'':real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2'' + t3'' = &1 - t1'':real
==> (inv ( &1 - t1''))*(t2'' + t3'') = (inv ( &1 - t1''))*( &1 - t1'':real)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C)= &1<=> &1- A*C = A*B`]
THEN STRIP_TAC
THEN MRESA_TAC REAL_LT_MUL[`inv(&1- t1'':real)`;`t3'':real`]
THEN MP_TAC(REAL_ARITH`&0<inv(&1- t1'':real)*(t3'':real)==> ~(inv(&1- t1'':real)*(t3'':real)= &0)`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0<inv(&1- t1'':real)==> ~(inv(&1- t1'':real)= &0)`)
THEN RESA_TAC
THEN ASSUME_TAC(REAL_ARITH`~(&1= &0)`)
THEN STRIP_TAC
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN DISCH_TAC
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v, w}= {(--x) +x, (--x) +v, (--x) +u, (--x) +w}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, w}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN DISCH_TAC
THEN MRESA_TAC continuous_coplanar_fan[`(vec 0):real^3 `;` u-x:real^3`;` v-x:real^3`;`  w-x:real^3`]
THEN POP_ASSUM (fun th-> MRESAL1_TAC th`(inv(&1- t1'')) * t3'':real`[VECTOR_ARITH`(A*B)%X+(A*C)%Y=A%(B%X+C%Y)`])
THEN MRESAL_TAC COPLANAR_SCALE_ALL [`&1`;`&1`;`inv(&1- t1'')`;`u - x:real^3`; `v - x:real^3`; `v'-x:real^3`]
[VECTOR_ARITH`&1 %A=A`]
THEN ASSUME_TAC(SET_RULE`IMAGE (\x'. --x + x') {x:real^3,  u, v,v'}= {(--x) +x, (--x) +u, (--x) +v, (--x) + v'}`)
THEN MRESAL_TAC COPLANAR_TRANSLATION_EQ[`--(x:real^3)`;`{x:real^3, u, v, v'}`][VECTOR_ARITH`(--A)+B=B-A:real^3`;VECTOR_ARITH`A-A=(vec 0):real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,B,C}`]
THEN DISCH_TAC
THEN REMOVE_THEN "EM"(fun th-> MRESA1_TAC th `v':real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN MRESA_TAC exists_element_in_half_sapace_fan
[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v':real^3`;`  v:real^3`;`u:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u':real^3)`;
` (v':real^3)`]
THEN MRESAL_TAC exists_cut_small_edges_fan_le[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`&1:real`;`v':real^3`;`u':real^3`]
[REAL_ARITH`&0< &1/\ &1<= &1`;VECTOR_ARITH`(&1 - &1) % u + &1 % w=w`]
THEN MRESAL_TAC not_cut_in_edges_fan 
[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`&1 -t:real`]
[REAL_ARITH`(&0< &1- t<=> t< &1)/\ (&1- t< &1<=> &0< t)`;VECTOR_ARITH`(&1 - (&1 - t)) % u + (&1 - t) % w=(&1 - t) % w +t % u`;SET_RULE`aff_gt {x} {v, (&1 - t) % w + t % u} INTER
      {v | ?e. e IN E /\ v IN aff_ge {x} e} =
      {}<=> !e. e IN E ==> aff_gt {x} {v, (&1 - t) % w + t % u} INTER
      aff_ge {x} e =
      {}`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{v',u':real^3}`)
THEN MRESA_TAC aff_gt_subset_aff_ge[`x:real^3`;`v':real^3`;`u':real^3`;]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];(*14*)



SUBGOAL_THEN`~(v' IN aff_gt {x} { x',v:real^3})` ASSUME_TAC
THENL(*15*)[
REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN MRESA_TAC aff_gt3_subset_aff_gt[`x:real^3`;`v:real^3`;`w:real^3`;`x':real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC 
THEN SET_TAC[];(*15*)

MP_TAC(ISPECL[`x:real^3`;`x':real^3`;`v:real^3`;`v':real^3`] decomposition_planar_by_angle_fan)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*16*)[
MRESA_TAC aff_gt12_subset_aff_ge[`x:real^3`;`v':real^3`;`w':real^3`;`x':real^3`]
THEN MP_TAC(SET_RULE`v IN aff_gt {x} {x', v':real^3} /\ aff_gt {x} {x', v'} SUBSET aff_ge {x} {v', w'}
==>v IN aff_ge {x} {v', w':real^3}`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`; `v:real^3`; `u:real^3`; `v':real^3`;`w':real^3`]properties_of_fan7)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*17*)[
 MRESA_TAC aff_gt2_subset_aff_ge [`x:real^3`;`v:real^3`; `w:real^3`;`x':real^3`]
THEN ASSUME_TAC(REAL_ARITH`pi <= pi`)
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`x':real^3`;`v:real^3`;`w:real^3`;`w':real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`A=A+B <=> B= &0`]
THEN DISCH_TAC
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`x':real^3`;`w:real^3`;`w':real^3`]
THEN MP_TAC(ISPECL[`x:real^3`;`x':real^3`;`w':real^3`;`w:real^3`] decomposition_planar_by_angle_fan)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*18*)[
MRESA_TAC aff_gt1_subset_aff_gt[`x:real^3`;`v:real^3`; `w:real^3`;`x':real^3`]
THEN ASM_TAC THEN SET_TAC[];(*18*)

MRESA_TAC aff_gt1_subset_aff_gt[`x:real^3`;`v:real^3`; `w:real^3`;`x':real^3`]
THEN ASM_TAC THEN SET_TAC[]](*18*);(*17*)

REMOVE_THEN "CUOI" MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN REWRITE_TAC[AZIM_REFL]
THEN MP_TAC(PI_WORKS)
THEN REAL_ARITH_TAC](*17*);(*16*)

POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN DISCH_TAC
THEN MRESA_TAC aff_ge_subset_aff_gt_union_aff_ge[`(x:real^3)`;`(v:real^3)`;`(x':real^3)`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[]
THEN DISCH_TAC
THEN MP_TAC(SET_RULE`aff_ge {x} {v, x':real^3} SUBSET aff_gt {x, v} {x'} UNION aff_ge {x} {v} /\ v' IN aff_ge {x} {v, x'} ==> v' IN aff_gt {x, v} {x'} UNION aff_ge {x} {v}`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC aff_gt_inter_aff_gt[`(x:real^3)`;`(x':real^3)`;`(v:real^3)`]
THEN ASM_REWRITE_TAC[INTER;IN_ELIM_THM;UNION]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MP_TAC(ISPECL[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`; `v':real^3`; `w':real^3`; `v:real^3`;`u:real^3`]properties1_of_fan7)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> MP_TAC(SYM(th)))
THEN DISCH_TAC
THEN MRESA_TAC point_in_aff_ge[`(x:real^3)`;`(v':real^3)`;`(w':real^3)`]
THEN MRESA_TAC aff_gt2_subset_aff_ge [`x:real^3`;`v:real^3`; `w:real^3`;`x':real^3`]
THEN ASSUME_TAC(REAL_ARITH`pi <= pi`)
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`x':real^3`;`v:real^3`;`w:real^3`;`w':real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`A=A+B <=> B= &0`]
THEN DISCH_TAC
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`x':real^3`;`w:real^3`;`w':real^3`]
THEN MP_TAC(ISPECL[`x:real^3`;`x':real^3`;`w':real^3`;`w:real^3`] decomposition_planar_by_angle_fan)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*17*)[
MRESA_TAC aff_gt1_subset_aff_gt[`x:real^3`;`v:real^3`; `w:real^3`;`x':real^3`]
THEN ASM_TAC THEN SET_TAC[];(*17*)

MRESA_TAC aff_ge1_subset_aff_ge[`x:real^3`;`v':real^3`; `w':real^3`;`x':real^3`]
THEN MP_TAC(SET_RULE`w IN aff_ge {x} { x',w':real^3} /\ aff_ge {x} { x',w'} SUBSET aff_ge {x} {v', w'}
==>w IN aff_ge {x} {v', w':real^3}`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`; `w:real^3`; `u:real^3`; `v':real^3`;`w':real^3`]properties_of_fan7)
THEN ASM_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN STRIP_TAC
THENL(*18*)[
POP_ASSUM MP_TAC 
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC 
THEN DISCH_THEN(LABEL_TAC "XONG")
THEN DISCH_TAC
THEN REMOVE_THEN "XONG" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC PI_WORKS
THEN REAL_ARITH_TAC;(*18*)

ASM_REWRITE_TAC[]](*18*)]]]]]]]]]]]]]]]]]);;








let DHVFGBC=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 a:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<a /\ a<= &1
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\  ~(aff_gt{x} {v,(&1-a)%u+ a % w} INTER {v | ?e. e IN E /\ v IN aff_ge {x} e}={})
==> a= &1 /\ independent {v - x, u - x, ((&1 - a) % u + a % w) - x}`,
MESON_TAC[independent_run_edges_fan;cut_in_edges_fan]);;



















(**********************************************************)
(**********************  RWXUYZZ  *************************)
(**********************************************************)

let exists_in_aff_gt=prove(`!x:real^3  v:real^3 u:real^3.
~collinear {x,v,u} ==> ?y:real^3. y IN aff_gt {x} {v, u}`,
REPEAT STRIP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&0 % x+ &1 / &2 % v+ &1/ &2 %u:real^3 `
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1/ &2`
THEN EXISTS_TAC`&1/ &2`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC);;

let cut_inside_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 w1:real^3.
~collinear {x,v,w1} /\ ~collinear {x,u,w} /\  ~collinear {x,v,u}
/\ ~collinear {x,v,w}
/\ &0< azim x u w v /\ azim x u w v < pi 
/\ &0< azim x v u w1 /\ azim x v u w1 < pi 
/\ &0< azim x v w1 w /\ azim x v w1 w < pi 
==> ~(aff_ge {x,v} {w1} INTER aff_gt {x} {u,w:real^3}={})`,
REWRITE_TAC[SET_RULE`~(A={})<=> ?x. x IN A`;IN_ELIM_THM; INTER] 
THEN REPEAT STRIP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`w1:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`u:real^3`;`w:real^3`]
THEN MRESAL_TAC  AFF_GE_2_1[`x:real^3`;`v:real^3`;`w1:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`u:real^3`;`w:real^3`][IN_ELIM_THM]
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`u:real^3`;`v:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"CON")
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`v:real^3`;`w1:real^3`;`u:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"CON BE")
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`v:real^3`;`w:real^3`;`w1:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"CON EM")
THEN ABBREV_TAC`a1=(v-x):real^3`
THEN ABBREV_TAC`a2=(w1-x):real^3`
THEN ABBREV_TAC`a3=(w-x) :real^3`
THEN ABBREV_TAC`a4=(u-x):real^3`
THEN ABBREV_TAC`va=a1 cross a2:real^3`
THEN ABBREV_TAC`vb=a3 cross a4:real^3`
THEN EXISTS_TAC`(vb:real^3) cross (va:real^3)+(x:real^3)`
THEN STRIP_TAC
THENL(*2*)[
EXISTS_TAC`&1-(vb:real^3) dot (a2:real^3)+ vb dot (a1:real^3)`
THEN EXISTS_TAC`(vb:real^3) dot (a2:real^3)`
THEN EXISTS_TAC`--((vb:real^3) dot (a1:real^3))`
THEN ASM_REWRITE_TAC[REAL_ARITH`(&1 - vb dot a2 + vb dot a1) + vb dot a2 + --(vb dot a1) = &1`]
THEN SUBGOAL_THEN `&0<= --((vb:real^3) dot (a1:real^3))` ASSUME_TAC
THENL(*3*)[
EXPAND_TAC"vb"
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[DOT_LNEG]
THEN REMOVE_THEN "CON"MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[] 
THEN REAL_ARITH_TAC;
ASM_REWRITE_TAC[]
THEN EXPAND_TAC"va"
THEN REWRITE_TAC[CROSS_LAGRANGE;VECTOR_MUL_LNEG]
THEN EXPAND_TAC"a1"
THEN EXPAND_TAC"a2"
THEN VECTOR_ARITH_TAC];(*2*)
ONCE_REWRITE_TAC[CROSS_SKEW]
THEN EXPAND_TAC"vb"
THEN REWRITE_TAC[CROSS_LAGRANGE;]
THEN EXISTS_TAC`&1+(va:real^3) dot (a4:real^3)- va dot (a3:real^3)`
THEN EXISTS_TAC`((va:real^3) dot (a3:real^3))`
THEN EXISTS_TAC`--(va:real^3) dot (a4:real^3)`
THEN ASM_REWRITE_TAC[DOT_LNEG;VECTOR_MUL_LNEG;REAL_ARITH`(&1 + va dot a4 - va dot a3) + va dot a3 + --(va dot a4) = &1`;]
THEN STRIP_TAC
THENL(*3*)[
EXPAND_TAC"va"
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[DOT_LNEG;CROSS_TRIPLE]
THEN ASM_REWRITE_TAC[] 
THEN ASM_TAC 
THEN REAL_ARITH_TAC;
EXPAND_TAC"a3"
THEN EXPAND_TAC"a4"
THEN REWRITE_TAC[VECTOR_ARITH`(&1+A-B)%X+B%U+ --(A%V)=X-(A%(V-X)-B%(U-X))`]
THEN VECTOR_ARITH_TAC]]);;



let exists_cut_in_edge_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 w1:real^3.
~collinear {x,v,w1} /\ ~collinear {x,u,w} /\  ~collinear {x,v,u}
/\ ~collinear {x,v,w}
/\ &0< azim x u w v /\ azim x u w v < pi 
/\ &0< azim x v u w1 /\ azim x v u w1 < pi 
/\ &0< azim x v w1 w /\ azim x v w1 w < pi 
==> ?a. &0< a /\ a< &1
/\ (&1-a) %u + a % w IN aff_ge {x,v} {w1:real^3}`,

REPEAT STRIP_TAC 
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`w1:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`u:real^3`;`w:real^3`]
THEN MRESAL_TAC  AFF_GE_2_1[`x:real^3`;`v:real^3`;`w1:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`u:real^3`;`w:real^3`][IN_ELIM_THM]
THEN MRESA_TAC cut_inside_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`;`w1:real^3`]
THEN POP_ASSUM MP_TAC
THEN GEN_REWRITE_TAC(LAND_CONV o DEPTH_CONV)[SET_RULE`~(A={})<=> ?x. x IN A`;INTER;]
THEN REWRITE_TAC[IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`inv(&1-t1')*t3'`
THEN MP_TAC(REAL_ARITH`&0< t2' /\ &0< t3' /\ t1'+t2'+t3'= &1==>  &0 < &1- t1' /\ ~(&1- t1' = &0)/\ t2'+t3'= &1- t1'`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ISPEC`&1- (t1':real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`&1- (t1':real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2' + t3' = &1 - t1':real
==> (inv ( &1 - t1'))*(t2' + t3') = (inv ( &1 - t1'))*( &1 - t1':real)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C)= &1<=>  &1 -A*C=A*B`;REAL_ARITH`A< &1 <=> &0< &1-A`]
THEN STRIP_TAC
THEN MRESA_TAC REAL_LT_MUL[`inv (&1- t1'):real`;`t2':real`;]
THEN MRESA_TAC REAL_LT_MUL[`inv (&1- t1'):real`;`t3':real`;]
THEN REWRITE_TAC[VECTOR_ARITH`(A*B)%X+(A*C)%Y=A%(B%X+C%Y)`; VECTOR_ARITH`A%(t2' % u + t3' % w)= A%((t1'%x +t2' % u + t3' % w) - t1' %x) :real^3`]
THEN FIND_ASSUM MP_TAC`x' = t1' % x + t2' % u + t3' % w:real^3`
THEN DISCH_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th)])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(t1 % x + t2 % v + t3 % w1) - t1' % x=(t1-t1') % x + t2 % v + t3 % w1`;
VECTOR_ARITH`A%(B+C+D)=A%B+A%C+A%D`;VECTOR_ARITH`A%B%C=(A*B)%C`]
THEN EXISTS_TAC`(inv (&1 - t1') * (t1 - t1')):real`
THEN EXISTS_TAC`(inv (&1 - t1') * t2):real`
THEN EXISTS_TAC`(inv (&1 - t1') * t3):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (&1 - t1') * (t1 - t1') + inv (&1 - t1') * t2 + inv (&1 - t1') * t3
=inv (&1 - t1') * ((t1 +t2 + t3)-t1')`]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC);;



let notcoplanar_imp_notcollinear_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3.
~coplanar {x,v,u,w}==> ~collinear {x,u,w} /\  ~collinear {x,v,u}
/\ ~collinear {x,v,w}`,
REPEAT GEN_TAC THEN STRIP_TAC
THEN MRESA_TAC NOT_COPLANAR_NOT_COLLINEAR [`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC NOT_COPLANAR_NOT_COLLINEAR [`x:real^3`;`u:real^3`;`w:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,B,C}`]
THEN RESA_TAC
THEN MRESAL_TAC NOT_COPLANAR_NOT_COLLINEAR [`x:real^3`;`v:real^3`;`w:real^3`;`u:real^3`][SET_RULE`{A,B,C,D}={A,B,D,C}`]);;

let properties_of_fully_surrounded1_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 w1:real^3.
~coplanar {x,v,u,w}/\ &0< azim x u w v /\ azim x u w v < pi 
==> &0 < azim x v u w /\ azim x v u w < pi`,
REPEAT STRIP_TAC
THEN MRESA_TAC azim[`(x:real^3)`;` (v:real^3)`;` (u:real^3)`;`(w:real^3)`]
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THENL[
 MP_TAC(REAL_ARITH` &0<= azim x v u (w:real^3) ==> azim x v u w = &0 \/ &0< azim x v u w`)
THEN RESA_TAC
THEN MRESA_TAC AZIM_EQ_0_PI_IMP_COPLANAR[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`];

MP_TAC(REAL_ARITH` (azim x v u w = pi) \/ (pi < azim x v u w) \/  azim x v u w< pi`)
THEN RESA_TAC
THENL[
MRESA_TAC AZIM_EQ_0_PI_IMP_COPLANAR[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`];

MP_TAC(REAL_ARITH`pi< azim x v u w /\ azim x v u w < &2 * pi ==> &0< azim x v u w - pi /\ azim x v u w - pi< pi`)
THEN RESA_TAC
THEN MRESAL1_TAC SIN_POS_PI`azim x v u (w:real^3) -pi`[SIN_SUB; SIN_PI; COS_PI;REAL_ARITH`&0< A * -- &1 -B * &0 <=> A < &0`]
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC AZIM_TRANSLATION[`-- x:real^3`;`x:real^3`;`v:real^3`;` u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`x-x= vec 0`;VECTOR_ARITH`(-- X)+A= A-X:real^3`]
THEN DISCH_TAC
THEN MRESA_TAC JBDNJJB[`(v-x):real^3`;`u-x:real^3`;`w-x:real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM COLLINEAR_3;]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN RESA_TAC
THEN STRIP_TAC
THEN MRESAL_TAC REAL_LT_LMUL_EQ[`(((v - x) cross (u - x)) dot (w - x:real^3))`;`&0`;`t:real`][REAL_ARITH`A * &0= &0`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[DOT_LNEG;CROSS_TRIPLE]
THEN MRESA1_TAC SIN_POS_PI`azim x u  (w:real^3) v`
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC AZIM_TRANSLATION[`-- x:real^3`;`x:real^3`;`u:real^3`;` w:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`x-x= vec 0`;VECTOR_ARITH`(-- X)+A= A-X:real^3`]
THEN DISCH_TAC
THEN MRESA_TAC JBDNJJB[`(u-x):real^3`;`w-x:real^3`;`v-x:real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM COLLINEAR_3;]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN RESA_TAC
THEN STRIP_TAC
THEN MRESAL_TAC REAL_LT_LMUL_EQ[`&0`;`(((u - x) cross (w - x)) dot (v - x:real^3))`;`t':real`][REAL_ARITH`A * &0= &0`]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC]])
;;





let in_aff_2_2_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3.
~coplanar {x,v,u,w}
==> 
(!t:real. &0< t /\ t< &1
==> (!t1:real t2:real t3:real. &0<t3  /\ t1+t2+t3= &1
==>t1 % x + t2 % v + t3 % ((&1 - t) % u + t % w) IN aff_gt {x,v} {u,w}))`,


REPEAT STRIP_TAC
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`w:real^3`]
THEN SUBGOAL_THEN `DISJOINT {x,v:real^3} {u,w:real^3}` ASSUME_TAC
THENL[
REWRITE_TAC[DISJOINT_SYM;SET_RULE`{v:real^3,w:real^3}= {v} UNION {w}`;DISJOINT_UNION]
THEN REWRITE_TAC[SET_RULE`{v} UNION {w}={v:real^3,w:real^3}`]
THEN ONCE_REWRITE_TAC[DISJOINT_SYM]
THEN ASM_REWRITE_TAC[];
MRESAL_TAC AFF_GT_2_2[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3*(&1-t):real`
THEN EXISTS_TAC`t3*(t):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`t1 +t2+  t3 * (&1 - t) + t3 * t = t1+t2+t3:real`;VECTOR_ARITH`t1 % x + t2 % v + t3 % ((&1 - t) % u + t % w) =
 t1 % x + t2 % v + (t3 * (&1 - t)) % u  + (t3 * t) % w:real^3`]
THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_TAC
THEN REAL_ARITH_TAC]);;


let inequality4_aim_in_convex_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 a:real.
~coplanar {x,v,u,w}/\ &0< azim x u w v /\ azim x u w v < pi 
/\ &0< a /\ a < &1
==> 
&0< azim x v u ((&1 - a) % u + a % w)
/\ azim x v u ((&1 - a) % u + a % w)< azim x v u w `,

REPEAT STRIP_TAC
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`w:real^3`]
THEN MRESA_TAC properties_of_fully_surrounded1_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`;`w1:real^3`]
THEN MRESA_TAC WEDGE_LUNE_GT[`x:real^3`;` v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM (fun th-> MP_TAC(SYM(th)))
THEN DISCH_TAC
THEN MRESA_TAC in_aff_2_2_fan[`x:real^3`;` v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th `a:real`)
THEN POP_ASSUM(fun th-> MRESAL_TAC th [`&0:real`;`&0`;`&1`][REAL_ARITH`&0< &1/\ &0+ &0 + &1 = &1`;wedge;IN_ELIM_THM])
THEN POP_ASSUM MP_TAC  THEN POP_ASSUM MP_TAC  THEN REDUCE_VECTOR_TAC
THEN DISCH_TAC THEN DISCH_TAC
THEN ASM_REWRITE_TAC[]);;

let condition_to_in_aff_gt_by_angle=prove(`!x:real^3 v:real^3 u:real^3 s1:real.
~collinear {x,v,u} /\  &0< (v - x) dot (u - x) /\ &0< s1 
/\ s1< atn ((norm ((v - x) cross (u - x))) * inv((v - x) dot (u - x)))
==>
sin s1 % e1_fan x v u + cos s1 % e3_fan x v u + x IN aff_gt {x} {v, u}`,
REPEAT STRIP_TAC
THEN ASSUME_TAC(ISPEC`(norm ((v - x) cross (u - x)) * inv ((v - x) dot (u - x))):real`ATN_BOUNDS)
THEN MP_TAC (REAL_ARITH`s1< atn ((norm ((v - x) cross (u - x))) * inv((v - x) dot (u - x)))
/\ atn ((norm ((v - x) cross (u - x))) * inv((v - x) dot (u - x))) < pi/ &2
==> s1< pi / &2`) THEN RESA_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESAL_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN REWRITE_TAC[VECTOR_ARITH`sin s1 % e1_fan x v u + cos s1 % e3_fan x v u + x =
     t1 % x + t2 % v + t3 % u
<=> sin s1 % e1_fan x v u + cos s1 % e3_fan x v u  =
     (t1- &1) % x + t2 % v + t3 % u`;e1_fan;e2_fan;e3_fan;CROSS_LMUL;CROSS_RMUL]
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[CROSS_LAGRANGE;VECTOR_ARITH`-- (A-B) = B-A:real^3`]
THEN SUBGOAL_THEN`~(norm((v:real^3)-(x:real^3))= &0)` ASSUME_TAC
THENL[
ASM_REWRITE_TAC[NORM_EQ_0;VECTOR_ARITH`v-x=vec 0<=> x=v`];
MP_TAC(ISPEC`norm((v:real^3)-(x:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; REAL_ARITH`(((A*B)*D)*D)=A*B*(D pow 2)`;]
THEN ONCE_REWRITE_TAC[VECTOR_ARITH`A%(B-C)=A%B-A%C`]
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`;DOT_SQUARE_NORM;REAL_ARITH`(A*B*C pow 2) * D pow 2=A*B*(C*D) pow 2`;REAL_ARITH`A* &1 pow 2=A`;NORM_MUL;REAL_ABS_INV;REAL_ABS_NORM;REAL_INV_MUL;REAL_INV_INV
; ]
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B*C) * D pow 2= A*C * D*(D*B)`;REAL_ARITH`A*B* &1= A*B`;VECTOR_ARITH`A-B+C%D-C%E=A-B+C%(D-E)`;VECTOR_ARITH`A-C%D+B%D=A+(B-C)%D`;
REAL_ARITH`A*B-(C*D*B)*E=B*(A-C*D*E)`]
THEN ONCE_REWRITE_TAC[GSYM CROSS_SKEW]
THEN SUBGOAL_THEN`~(norm((v - x) cross (u - x:real^3))= &0)` ASSUME_TAC
THENL[
ASM_REWRITE_TAC[NORM_EQ_0]
THEN MP_TAC(ISPECL[`v-x:real^3`;`u-x:real^3`]CROSS_EQ_0)
THEN ONCE_REWRITE_TAC[GSYM COLLINEAR_3;]
THEN RESA_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN ASM_REWRITE_TAC[];
MP_TAC(ISPEC`norm((v - x) cross (u - x:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`(v - x) cross (u - x:real^3)`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(norm ((v - x) cross (u - x:real^3)) = &0)/\ &0 <= norm ((v - x) cross (u - x:real^3))==> &0< norm ((v - x) cross (u - x:real^3)) `)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`norm((v - x) cross (u - x:real^3))`

THEN ASSUME_TAC(ISPEC`(v - x:real^3)`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(norm ((v - x:real^3)) = &0)/\ &0 <= norm ((v - x:real^3))==> &0< norm ((v - x:real^3)) `)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`norm((v - x:real^3))`
THEN MRESA1_TAC COS_POS_PI2`s1:real`
THEN MRESA1_TAC SIN_POS_PI2`s1:real`
THEN EXISTS_TAC`&1-(sin s1 * norm (v - x) * inv (norm ((v - x) cross (u - x))))
-(inv (norm (v - x)) *
      (cos s1 -
       sin s1 * inv (norm ((v - x) cross (u - x))) * ((v - x) dot (u - x:real^3))))`
THEN EXISTS_TAC `(inv (norm (v - x)) *
      (cos s1 -
       sin s1 * inv (norm ((v - x) cross (u - x))) * ((v - x) dot (u - x:real^3))))`
THEN EXISTS_TAC`(sin s1 * norm (v - x) * inv (norm ((v - x) cross (u - x:real^3))))`
THEN STRIP_TAC
THENL[
MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<A-B<=> B<A`]
THEN ASSUME_TAC(PI_WORKS)
THEN MP_TAC(REAL_ARITH`&0< pi /\ &0< s1 ==> --(pi / &2) < s1`)
THEN RESA_TAC
THEN MRESAL_TAC  TAN_MONO_LT[`s1:real`;`atn (norm ((v - x) cross (u - x)) * inv ((v - x) dot (u - x))):real`][ATN_TAN]
THEN MRESAL_TAC REAL_LT_LMUL[`inv (norm ((v - x) cross (u - x:real^3)))`;`tan s1:real`;`norm ((v - x) cross (u - x)) * inv ((v - x) dot (u - x:real^3))`][REAL_ARITH`A*B*C=(A*B)*C`;REAL_ARITH`&1*A=A`]
THEN MP_TAC(REAL_ARITH`&0<(v - x) dot (u - x:real^3)==> ~((v - x) dot (u - x:real^3)= &0)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(v - x) dot (u - x:real^3)`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0<cos s1 ==> ~(cos s1= &0)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`cos s1`REAL_MUL_LINV)
THEN RESA_TAC
THEN MRESAL_TAC REAL_LT_RMUL[`inv (norm ((v - x) cross (u - x:real^3)))* tan s1:real`;`inv ((v - x) dot (u - x:real^3))`;`(v - x) dot (u - x:real^3)`][REAL_ARITH`(A*B)*C=A*C*B`;tan]

THEN MRESAL_TAC REAL_LT_RMUL[`inv (norm ((v - x) cross (u - x:real^3)))* ((v - x) dot (u - x:real^3))* sin s1 / cos s1`;`&1`;`cos s1`][REAL_ARITH`&1* A=A`;real_div;REAL_ARITH`(A*B*C*D)*E=(C*A*B)*(D*E)`;REAL_ARITH`A* &1=A`]
THEN ASM_TAC THEN REAL_ARITH_TAC;
STRIP_TAC
THENL[
MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[];

STRIP_TAC
THENL[
REAL_ARITH_TAC;
VECTOR_ARITH_TAC]]]]]);;







let condition1_to_in_aff_gt_by_angle=prove(`!x:real^3 v:real^3 u:real^3 s1:real.
~collinear {x,v,u} /\ &0< s1 /\ s1< pi/ &2
/\ (v - x) dot (u - x:real^3) <= &0
==>
sin s1 % e1_fan x v u + cos s1 % e3_fan x v u + x IN aff_gt {x} {v, u}`,

REPEAT STRIP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESAL_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN REWRITE_TAC[VECTOR_ARITH`sin s1 % e1_fan x v u + cos s1 % e3_fan x v u + x =
     t1 % x + t2 % v + t3 % u
<=> sin s1 % e1_fan x v u + cos s1 % e3_fan x v u  =
     (t1- &1) % x + t2 % v + t3 % u`;e1_fan;e2_fan;e3_fan;CROSS_LMUL;CROSS_RMUL]
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[CROSS_LAGRANGE;VECTOR_ARITH`-- (A-B) = B-A:real^3`]
THEN SUBGOAL_THEN`~(norm((v:real^3)-(x:real^3))= &0)` ASSUME_TAC
THENL[
ASM_REWRITE_TAC[NORM_EQ_0;VECTOR_ARITH`v-x=vec 0<=> x=v`];
MP_TAC(ISPEC`norm((v:real^3)-(x:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; REAL_ARITH`(((A*B)*D)*D)=A*B*(D pow 2)`;]
THEN ONCE_REWRITE_TAC[VECTOR_ARITH`A%(B-C)=A%B-A%C`]
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`;DOT_SQUARE_NORM;REAL_ARITH`(A*B*C pow 2) * D pow 2=A*B*(C*D) pow 2`;REAL_ARITH`A* &1 pow 2=A`;NORM_MUL;REAL_ABS_INV;REAL_ABS_NORM;REAL_INV_MUL;REAL_INV_INV
; ]
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B*C) * D pow 2= A*C * D*(D*B)`;REAL_ARITH`A*B* &1= A*B`;VECTOR_ARITH`A-B+C%D-C%E=A-B+C%(D-E)`;VECTOR_ARITH`A-C%D+B%D=A+(B-C)%D`;
REAL_ARITH`A*B-(C*D*B)*E=B*(A-C*D*E)`]
THEN ONCE_REWRITE_TAC[GSYM CROSS_SKEW]
THEN SUBGOAL_THEN`~(norm((v - x) cross (u - x:real^3))= &0)` ASSUME_TAC
THENL[
ASM_REWRITE_TAC[NORM_EQ_0]
THEN MP_TAC(ISPECL[`v-x:real^3`;`u-x:real^3`]CROSS_EQ_0)
THEN ONCE_REWRITE_TAC[GSYM COLLINEAR_3;]
THEN RESA_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN ASM_REWRITE_TAC[];
MP_TAC(ISPEC`norm((v - x) cross (u - x:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`(v - x) cross (u - x:real^3)`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(norm ((v - x) cross (u - x:real^3)) = &0)/\ &0 <= norm ((v - x) cross (u - x:real^3))==> &0< norm ((v - x) cross (u - x:real^3)) `)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`norm((v - x) cross (u - x:real^3))`

THEN ASSUME_TAC(ISPEC`(v - x:real^3)`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(norm ((v - x:real^3)) = &0)/\ &0 <= norm ((v - x:real^3))==> &0< norm ((v - x:real^3)) `)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`norm((v - x:real^3))`
THEN MRESA1_TAC COS_POS_PI2`s1:real`
THEN MRESA1_TAC SIN_POS_PI2`s1:real`
THEN EXISTS_TAC`&1-(sin s1 * norm (v - x) * inv (norm ((v - x) cross (u - x))))
-(inv (norm (v - x)) *
      (cos s1 -
       sin s1 * inv (norm ((v - x) cross (u - x))) * ((v - x) dot (u - x:real^3))))`
THEN EXISTS_TAC `(inv (norm (v - x)) *
      (cos s1 -
       sin s1 * inv (norm ((v - x) cross (u - x))) * ((v - x) dot (u - x:real^3))))`
THEN EXISTS_TAC`(sin s1 * norm (v - x) * inv (norm ((v - x) cross (u - x:real^3))))`
THEN STRIP_TAC
THENL[
MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<A-B<=> B<A`]
THEN MATCH_MP_TAC(REAL_ARITH`&0< cos s1 /\ sin s1 * inv (norm ((v - x) cross (u - x))) * ((v - x) dot (u - x)) <= &0==> sin s1 * inv (norm ((v - x) cross (u - x))) * ((v - x) dot (u - x)) < cos s1`)
THEN ASM_REWRITE_TAC[REAL_ARITH`A *B *C<= &0<=> &0<= A*B*(-- C)`]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN MP_TAC(REAL_ARITH`&0< sin s1==> &0<= sin s1`) THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN MP_TAC(REAL_ARITH`&0< inv (norm ((v - x) cross (u - x)))==> &0<= inv (norm ((v - x) cross (u - x)))`) THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC;
STRIP_TAC
THENL[
MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[];
STRIP_TAC
THENL[
REAL_ARITH_TAC;
VECTOR_ARITH_TAC]]]]]);;



let angle_is_small_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ fan80(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
==>
 azim x v u w <= azim x v u (sigma_fan x V E v u) `,


REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "BE")
THEN DISCH_THEN(LABEL_TAC"EM")
THEN USE_THEN "BE" MP_TAC
THEN REWRITE_TAC[fan80]
THEN DISCH_TAC
THEN POP_ASSUM (fun th->   MRESA_TAC th [`u:real^3`;`w:real^3`]
THEN MRESA_TAC th [`v:real^3`;`u:real^3`])
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`]
THEN REMOVE_THEN "EM" (fun th-> MRESA1_TAC th `v:real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN DISJ_CASES_TAC(SET_RULE`set_of_edge v V E = {u} \/ ~(set_of_edge v V E = {u:real^3})`)
THENL(*1*)[
MRESA_TAC CARD_SING[`u:real^3`; `(set_of_edge v V E):real^3->bool`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th->REWRITE_TAC[SYM(th)])
THEN ASM_TAC THEN ARITH_TAC;(*1*)
MRESA_TAC SIGMA_FAN[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;` (u:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` sigma_fan x V E v (u:real^3)`;` (v:real^3)`]
THEN POP_ASSUM MP_TAC THEN RESA_TAC  
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`w:real^3`]
THEN  ABBREV_TAC`w1=(sigma_fan x V E v u) :real^3`
THEN MRESA_TAC properties_of_fully_surrounded1_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`;`w1:real^3`]
THEN DISJ_CASES_TAC(REAL_ARITH`  azim x v u w1 < azim x v u w\/ azim x v u w <= azim x v u w1 `)
THENL(*2*)[
MP_TAC(REAL_ARITH`azim x v u w1 < azim x v u w ==> azim x v u w1 <= azim x v u w`)
THEN RESA_TAC
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w1:real^3`;`w:real^3`]
THEN MP_TAC(REAL_ARITH` &0< azim x v u w1 /\ azim x v u w1 < azim x v u w /\ azim x v u w = azim x v u w1 + azim x v w1 w
==> &0< azim x v w1 w /\ azim x v w1 w <= azim x v u w`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`azim x v w1 w <= azim x v u w /\ azim x v u w < pi
==> azim x v w1 w < pi`)
THEN RESA_TAC
THEN MRESA_TAC exists_cut_in_edge_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`;`w1:real^3`]
THEN POP_ASSUM MP_TAC
THEN GEN_REWRITE_TAC(LAND_CONV o DEPTH_CONV)[SET_RULE`~(A={})<=> ?x. x IN A`;INTER]
THEN REWRITE_TAC[IN_ELIM_THM] 
THEN STRIP_TAC
THEN ABBREV_TAC`va=(&1-a)%u + a%w:real^3`
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`]
THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w1:real^3`;`va:real^3`] decomposition_planar_by_angle_fan)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*3*)[
MRESA_TAC point_in_aff_ge[`(x:real^3)`;`(v:real^3)`;`(w1:real^3)`]
THEN MRESA_TAC not_cut_in_edges_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`]
THEN MP_TAC(SET_RULE`w1 IN aff_gt {x} {v, va}
 /\ w1 IN aff_ge {x} {v, w1}==> ~(aff_gt {x} {v, va} INTER aff_ge {x:real^3} {v, w1}={})`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`{v,w1:real^3} IN E`
THEN SET_TAC[];(*3*)
MRESA_TAC pos_in_aff_ge_fan [`x:real^3`;`u:real^3`;`w:real^3`;`a:real`]
THEN MP_TAC(SET_RULE`va IN aff_ge {x} {v, w1:real^3} /\ va IN aff_ge {x} {u, w} 
==>va IN  aff_ge {x} {u, w} INTER aff_ge {x} {v, w1:real^3} `)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC `FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN;fan7]
THEN STRIP_TAC
THEN POP_ASSUM  (fun th-> MP_TAC(ISPECL[`{(u:real^3),(w:real^3)}`;`{(v:real^3),(w1:real^3)}`]th)) 
  THEN ASM_REWRITE_TAC[UNION;IN_ELIM_THM;]
THEN DISCH_TAC THEN ASM_REWRITE_TAC[]
THEN MP_TAC(SET_RULE`~(v=u) /\ DISJOINT{x,v} {w} ==> ~(v IN {u,w:real^3}) `)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(v IN {u,w:real^3})/\ ~(u=w)==> ~({u, w} INTER {v, w1} = {u, w:real^3})`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~({u, w} INTER {v, w1} = {u, w:real^3}) ==> {u, w} INTER {v, w1} PSUBSET {u, w} `)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`{u, w} INTER {v, w1} PSUBSET {u, w:real^3} ==> {u, w} INTER {v, w1} SUBSET {u}\/  {u, w} INTER {v, w1} SUBSET {w} `)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*4*)[
MRESAL_TAC AFF_GE_MONO_RIGHT [`{x:real^3}`;`{u, w} INTER {v, w1:real^3}`; `{u:real^3}`][DISJOINT;SET_RULE`{x} INTER {u} = {}<=> ~(x=u:real^3)`]
THEN STRIP_TAC
THEN MRESAL_TAC AFF_GE_SUBSET_AFFINE_HULL[`{x:real^3}`;`{u:real^3}`][SET_RULE`{x} UNION {u}={x,u}`;GSYM aff]
THEN MP_TAC(SET_RULE`aff_ge {x:real^3} ({u, w} INTER {v, w1}) SUBSET aff_ge {x} {u}/\ aff_ge {x} {u} SUBSET aff {x, u}/\ va IN aff_ge {x} ({u, w} INTER {v, w1})==> va IN aff {x,u}`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC"va"
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - a) % u + a % w = u' % x + v'' % u
<=> a % w = u' % x + (v''+a- &1) % u`]
THEN MP_TAC(REAL_ARITH`&0< a==> ~(a = &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`a:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`a % w = u' % x + (v' + a - &1) % u:real^3 ==> (inv (a))%(a % w ) = (inv (a))%( u' % x + (v' + a - &1) % u)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN `w IN aff {x,u:real^3}` ASSUME_TAC
THENL(*5*)[
REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN EXISTS_TAC`(inv a * u'):real`
THEN EXISTS_TAC`(inv a * (v' + a - &1)):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv a * u' + inv a * (v' + a - &1) =inv a * ( a+ (u' + v') - &1)`;REAL_ARITH`A+ &1- &1=A`];(*5*)
POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC `~(w IN aff {x, u:real^3})`
THEN SET_TAC[]](*5*);(*4*)

FIND_ASSUM MP_TAC`{u,w:real^3} IN E`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN STRIP_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` u:real^3`;`w:real^3`]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN MRESAL_TAC AFF_GE_MONO_RIGHT [`{x:real^3}`;`{u, w} INTER {v, w1:real^3}`; `{w:real^3}`][DISJOINT;SET_RULE`{x} INTER {w} = {}<=> ~(x=w:real^3)`]
THEN STRIP_TAC
THEN MRESAL_TAC AFF_GE_SUBSET_AFFINE_HULL[`{x:real^3}`;`{w:real^3}`][SET_RULE`{x} UNION {w}={x,w}`;GSYM aff]
THEN MP_TAC(SET_RULE`aff_ge {x:real^3} ({u, w} INTER {v, w1}) SUBSET aff_ge {x} {w}/\ aff_ge {x} {w} SUBSET aff {x, w}/\ va IN aff_ge {x} ({u, w} INTER {v, w1})==> va IN aff {x,w}`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC"va"
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - a) % u + a % w = u' % x + v'' % w
<=> (&1-a) % u = u' % x + (v''-a) % w`]
THEN MP_TAC(REAL_ARITH`a < &1==> ~(&1- a= &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`&1-a:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(&1-a) % u = u' % x + (v' - a) % w:real^3 ==> (inv (&1-a))%((&1-a) % u ) = (inv (&1-a))%( u' % x + (v' - a) % w)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN `u IN aff {x,w:real^3}` ASSUME_TAC
THENL(*5*)[
REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN EXISTS_TAC`(inv (&1-a) * u'):real`
THEN EXISTS_TAC`(inv (&1-a) * (v' - a)):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (&1-a') * u' + inv (&1-a') * (v' - a') =inv (&1-a') * ( (u' + v') - a')`;];(*5*)

POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC `~(u IN aff {x, w:real^3})`
THEN SET_TAC[]](*5*)](*4*)](*3*);(*2*)

ASM_REWRITE_TAC[]]]);;


let angle_is_smallpi_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ fan80(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
==>
&0< azim x v u w /\ azim x v u w <pi`,
REPEAT STRIP_TAC
THENL[ 
POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "BE")
THEN DISCH_THEN(LABEL_TAC"EM")
THEN USE_THEN "BE" MP_TAC
THEN REWRITE_TAC[fan80]
THEN DISCH_TAC
THEN POP_ASSUM (fun th->   MRESA_TAC th [`u:real^3`;`w:real^3`]
THEN MRESA_TAC th [`v:real^3`;`u:real^3`])
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN SUBGOAL_THEN`~(azim x v u (w:real^3)= &0)`ASSUME_TAC
THENL[STRIP_TAC
THEN MRESA_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`];
MATCH_MP_TAC(REAL_ARITH`~(azim x v u (w:real^3)= &0)/\ &0<= azim x v u w ==> &0 < azim x v u w`)
THEN ASM_REWRITE_TAC[azim]];

POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "BE")
THEN DISCH_THEN(LABEL_TAC"EM")
THEN USE_THEN "BE" MP_TAC
THEN REWRITE_TAC[fan80]
THEN DISCH_TAC
THEN POP_ASSUM (fun th->   MRESA_TAC th [`u:real^3`;`w:real^3`]
THEN MRESA_TAC th [`v:real^3`;`u:real^3`])
THEN MRESA_TAC angle_is_small_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;` u:real^3`;`w:real^3`]
THEN ASM_TAC THEN REAL_ARITH_TAC]);;



let exists_rw_dart_inter_aff_gt_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 .
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ fan80(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
==>
(?h:real. &0< h /\
(!t:real. &0< t /\ t< h==> 
(!s:real. &0<s /\ s< pi/ &2
==>
~(rw_dart_fan x V E ((x:real^3),(v:real^3),(u:real^3),(sigma_fan x V E v (u:real^3))) (cos(s)) INTER aff_gt {x} {v, (&1-t)%u+t%w}={})
)))`,


REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "BE")
THEN DISCH_THEN(LABEL_TAC"EM")
THEN USE_THEN "BE" MP_TAC
THEN REWRITE_TAC[fan80]
THEN DISCH_TAC
THEN POP_ASSUM (fun th->   MRESA_TAC th [`u:real^3`;`w:real^3`]
THEN MRESA_TAC th [`v:real^3`;`u:real^3`])
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`]
THEN REMOVE_THEN "EM" (fun th-> MRESA1_TAC th `v:real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN DISJ_CASES_TAC(SET_RULE`set_of_edge v V E = {u} \/ ~(set_of_edge v V E = {u:real^3})`)
THENL(*1*)[
MRESA_TAC CARD_SING[`u:real^3`; `(set_of_edge v V E):real^3->bool`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th->REWRITE_TAC[SYM(th)])
THEN ASM_TAC THEN ARITH_TAC;(*1*)

MRESA_TAC SIGMA_FAN[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;` (u:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` sigma_fan x V E v (u:real^3)`;` (v:real^3)`]
THEN POP_ASSUM MP_TAC THEN RESA_TAC  
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`w:real^3`]
THEN  ABBREV_TAC`w1=(sigma_fan x V E v u) :real^3`
THEN MRESA_TAC properties_of_fully_surrounded1_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`;`w1:real^3`]
THEN DISJ_CASES_TAC(REAL_ARITH`  azim x v u w1 < azim x v u w\/ azim x v u w <= azim x v u w1 `)
THENL(*2*)[
MP_TAC(REAL_ARITH`azim x v u w1 < azim x v u w ==> azim x v u w1 <= azim x v u w`)
THEN RESA_TAC
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w1:real^3`;`w:real^3`]
THEN MP_TAC(REAL_ARITH` &0< azim x v u w1 /\ azim x v u w1 < azim x v u w /\ azim x v u w = azim x v u w1 + azim x v w1 w
==> &0< azim x v w1 w /\ azim x v w1 w <= azim x v u w`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`azim x v w1 w <= azim x v u w /\ azim x v u w < pi
==> azim x v w1 w < pi`)
THEN RESA_TAC
THEN MRESA_TAC exists_cut_in_edge_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`;`w1:real^3`]
THEN POP_ASSUM MP_TAC
THEN GEN_REWRITE_TAC(LAND_CONV o DEPTH_CONV)[SET_RULE`~(A={})<=> ?x. x IN A`;INTER]
THEN REWRITE_TAC[IN_ELIM_THM] 
THEN STRIP_TAC
THEN ABBREV_TAC`va=(&1-a)%u + a%w:real^3`
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`]
THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w1:real^3`;`va:real^3`] decomposition_planar_by_angle_fan)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*3*)[
MRESA_TAC point_in_aff_ge[`(x:real^3)`;`(v:real^3)`;`(w1:real^3)`]
THEN MRESA_TAC not_cut_in_edges_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`a:real`]
THEN MP_TAC(SET_RULE`w1 IN aff_gt {x} {v, va}
 /\ w1 IN aff_ge {x} {v, w1}==> ~(aff_gt {x} {v, va} INTER aff_ge {x:real^3} {v, w1}={})`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`{v,w1:real^3} IN E`
THEN SET_TAC[];(*3*)

MRESA_TAC pos_in_aff_ge_fan [`x:real^3`;`u:real^3`;`w:real^3`;`a:real`]
THEN MP_TAC(SET_RULE`va IN aff_ge {x} {v, w1:real^3} /\ va IN aff_ge {x} {u, w} 
==>va IN  aff_ge {x} {u, w} INTER aff_ge {x} {v, w1:real^3} `)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC `FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN;fan7]
THEN STRIP_TAC
THEN POP_ASSUM  (fun th-> MP_TAC(ISPECL[`{(u:real^3),(w:real^3)}`;`{(v:real^3),(w1:real^3)}`]th)) 
  THEN ASM_REWRITE_TAC[UNION;IN_ELIM_THM;]
THEN DISCH_TAC THEN ASM_REWRITE_TAC[]
THEN MP_TAC(SET_RULE`~(v=u) /\ DISJOINT{x,v} {w} ==> ~(v IN {u,w:real^3}) `)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(v IN {u,w:real^3})/\ ~(u=w)==> ~({u, w} INTER {v, w1} = {u, w:real^3})`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~({u, w} INTER {v, w1} = {u, w:real^3}) ==> {u, w} INTER {v, w1} PSUBSET {u, w} `)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`{u, w} INTER {v, w1} PSUBSET {u, w:real^3} ==> {u, w} INTER {v, w1} SUBSET {u}\/  {u, w} INTER {v, w1} SUBSET {w} `)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL(*4*)[
MRESAL_TAC AFF_GE_MONO_RIGHT [`{x:real^3}`;`{u, w} INTER {v, w1:real^3}`; `{u:real^3}`][DISJOINT;SET_RULE`{x} INTER {u} = {}<=> ~(x=u:real^3)`]
THEN STRIP_TAC
THEN MRESAL_TAC AFF_GE_SUBSET_AFFINE_HULL[`{x:real^3}`;`{u:real^3}`][SET_RULE`{x} UNION {u}={x,u}`;GSYM aff]
THEN MP_TAC(SET_RULE`aff_ge {x:real^3} ({u, w} INTER {v, w1}) SUBSET aff_ge {x} {u}/\ aff_ge {x} {u} SUBSET aff {x, u}/\ va IN aff_ge {x} ({u, w} INTER {v, w1})==> va IN aff {x,u}`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC"va"
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - a) % u + a % w = u' % x + v'' % u
<=> a % w = u' % x + (v''+a- &1) % u`]
THEN MP_TAC(REAL_ARITH`&0< a==> ~(a = &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`a:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`a % w = u' % x + (v' + a - &1) % u:real^3 ==> (inv (a))%(a % w ) = (inv (a))%( u' % x + (v' + a - &1) % u)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN `w IN aff {x,u:real^3}` ASSUME_TAC
THENL(*5*)[
REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN EXISTS_TAC`(inv a * u'):real`
THEN EXISTS_TAC`(inv a * (v' + a - &1)):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv a * u' + inv a * (v' + a - &1) =inv a * ( a+ (u' + v') - &1)`;REAL_ARITH`A+ &1- &1=A`];(*5*)
POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC `~(w IN aff {x, u:real^3})`
THEN SET_TAC[]](*5*);(*4*)

FIND_ASSUM MP_TAC`{u,w:real^3} IN E`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN STRIP_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` u:real^3`;`w:real^3`]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN MRESAL_TAC AFF_GE_MONO_RIGHT [`{x:real^3}`;`{u, w} INTER {v, w1:real^3}`; `{w:real^3}`][DISJOINT;SET_RULE`{x} INTER {w} = {}<=> ~(x=w:real^3)`]
THEN STRIP_TAC
THEN MRESAL_TAC AFF_GE_SUBSET_AFFINE_HULL[`{x:real^3}`;`{w:real^3}`][SET_RULE`{x} UNION {w}={x,w}`;GSYM aff]
THEN MP_TAC(SET_RULE`aff_ge {x:real^3} ({u, w} INTER {v, w1}) SUBSET aff_ge {x} {w}/\ aff_ge {x} {w} SUBSET aff {x, w}/\ va IN aff_ge {x} ({u, w} INTER {v, w1})==> va IN aff {x,w}`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC"va"
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - a) % u + a % w = u' % x + v'' % w
<=> (&1-a) % u = u' % x + (v''-a) % w`]
THEN MP_TAC(REAL_ARITH`a < &1==> ~(&1- a= &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`&1-a:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(&1-a) % u = u' % x + (v' - a) % w:real^3 ==> (inv (&1-a))%((&1-a) % u ) = (inv (&1-a))%( u' % x + (v' - a) % w)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN `u IN aff {x,w:real^3}` ASSUME_TAC
THENL(*5*)[REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN EXISTS_TAC`(inv (&1-a) * u'):real`
THEN EXISTS_TAC`(inv (&1-a) * (v' - a)):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (&1-a') * u' + inv (&1-a') * (v' - a') =inv (&1-a') * ( (u' + v') - a')`;];(*5*)

POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC `~(u IN aff {x, w:real^3})`
THEN SET_TAC[]](*5*)](*4*)](*3*);(*2*)

EXISTS_TAC`&1`
THEN REWRITE_TAC[REAL_ARITH`&0< &1`]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ABBREV_TAC`va=(&1-t)%u + t%w:real^3`
THEN MRESA_TAC not_collinear_is_properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`t:real`]
THEN MRESA_TAC inequality4_aim_in_convex_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`;`t:real`]
THEN ASM_REWRITE_TAC[rw_dart_fan;w_dart_fan]
THEN SUBGOAL_THEN`aff_gt {x} {v,va:real^3} SUBSET wedge x v u w1` ASSUME_TAC
THENL(*3*)[
REWRITE_TAC[SUBSET; IN_ELIM_THM; wedge]
THEN GEN_TAC
THEN MRESAL_TAC aff_gt_inter_aff_gt [`(x:real^3)`;`(v:real^3)`;`(va:real^3)`][INTER;IN_ELIM_THM]
THEN STRIP_TAC
THEN MRESAL_TAC properties_of_collinear4_points_fan[`x:real^3`;`v:real^3`;`va:real^3`;`x':real^3`][IN_ELIM_THM]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC AZIM_EQ_ALT[`x:real^3`;`v:real^3`;`u:real^3`; `x':real^3`;`va:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN RESA_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;(*3*)

MP_TAC(SET_RULE`aff_gt {x} {v, va:real^3} SUBSET wedge x v u w1 ==> (wedge x v u w1 INTER rcone_fan x v (cos s)) INTER aff_gt {x} {v, va}=(rcone_fan x v (cos s)) INTER aff_gt {x} {v, va}`)
THEN RESA_TAC
THEN REWRITE_TAC[rcone_fan;INTER;IN_ELIM_THM]
THEN REWRITE_TAC[SET_RULE`~(A={})<=> ?y. y IN A`;IN_ELIM_THM]
THEN DISJ_CASES_TAC(REAL_ARITH`(v - x) dot (va - x:real^3) <= &0 \/ &0< (v - x) dot (va - x)`)
THENL(*4*)[
ABBREV_TAC`s1= s/ &2:real`
THEN MP_TAC(REAL_ARITH`&0< s /\ s< pi/ &2/\ s1= s/ &2 /\ &0< pi ==> &0<= s1 /\ s1< s /\  s <= pi/\ &0<s1 /\ s1<pi/ &2`)
THEN REWRITE_TAC[PI_WORKS]
THEN RESA_TAC
THEN EXISTS_TAC`sin (s1) % (e1_fan x v va) + (cos s1) %(e3_fan x v va)+x :real^3 `
THEN REWRITE_TAC[dist;vector_norm;VECTOR_ARITH`(B+C+A)-A=(B+C:real^3)`; DOT_LADD;DOT_RADD;DOT_RMUL;DOT_LMUL;]
THEN MRESAL_TAC properties_coordinate[`x:real^3`;`v:real^3`;`va:real^3`][orthonormal]
THEN ONCE_REWRITE_TAC[DOT_SYM]
THEN ASSUME_TAC(ISPEC`s1:real`SIN_CIRCLE)
THEN ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`sin s * sin s + cos s * cos s =(sin s) pow 2 + (cos s) pow 2`;e3_fan;DOT_RMUL; ]
THEN ONCE_REWRITE_TAC[GSYM vector_norm;]
THEN REWRITE_TAC[DOT_SQUARE_NORM;]
THEN MRESAL1_TAC  SQRT_POW_2`&1`[REAL_ARITH`a pow 2 = a * a`;REAL_ARITH`&0<= &1`;]
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC SQRT_MUL[`&1`;`&1`][REAL_ARITH`&0<= &1`;]
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th); REAL_ARITH`&1* &1= &1`])
THEN RESA_TAC
THEN REDUCE_ARITH_TAC
THEN SUBGOAL_THEN`~(norm((v:real^3)-(x:real^3))= &0)` ASSUME_TAC
THENL(*5*)[ ASM_REWRITE_TAC[NORM_EQ_0;VECTOR_ARITH`v-x=vec 0<=> x=v`];(*5*)

MP_TAC(ISPEC`norm((v:real^3)-(x:real^3))`REAL_MUL_LINV)
THEN REWRITE_TAC[REAL_ARITH`A*B*D *D=(B*D) *D *A`;REAL_ARITH`A>B<=> B<A`]
THEN RESA_TAC
THEN REDUCE_ARITH_TAC
THEN ASSUME_TAC(ISPEC`v-x:real^3`NORM_POS_LE)
THEN STRIP_TAC
THENL(*6*)[
MATCH_MP_TAC REAL_LT_LMUL
THEN MP_TAC(REAL_ARITH`~(norm (v - x:real^3) = &0)/\ &0 <= norm (v - x)==> &0< norm (v - x) `)
THEN RESA_TAC
THEN MATCH_MP_TAC COS_MONO_LT
THEN ASM_REWRITE_TAC[];(*6*)

MRESA_TAC condition1_to_in_aff_gt_by_angle[`x:real^3`;`v:real^3`;`va:real^3`;`s1:real`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[e3_fan]](*6*)](*5*);(*4*)
SUBGOAL_THEN`&0<(atn ((norm ((v - x) cross (va - x))) * inv((v - x) dot (va - x))))` ASSUME_TAC
THENL(*5*)[
MP_TAC(ISPEC`(v - x) dot (va - x:real^3)`REAL_LT_INV)
THEN RESA_TAC
THEN ASSUME_TAC(PI_WORKS)
THEN MP_TAC(REAL_ARITH`&0< pi ==> --(pi / &2) < &0`)
THEN RESA_TAC
THEN MRESAL_TAC  ATN_MONO_LT[`&0:real`;` (norm ((v - x) cross (va - x)) * inv ((v - x) dot (va - x))):real`][ ATN_0]
THEN POP_ASSUM MATCH_MP_TAC
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN SUBGOAL_THEN`~(norm((v - x) cross (va - x:real^3))= &0)` ASSUME_TAC
THENL(*6*)[ASM_REWRITE_TAC[NORM_EQ_0]
THEN MP_TAC(ISPECL[`v-x:real^3`;`va-x:real^3`]CROSS_EQ_0)
THEN ONCE_REWRITE_TAC[GSYM COLLINEAR_3;]
THEN RESA_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN ASM_REWRITE_TAC[];
POP_ASSUM MP_TAC
THEN MP_TAC(ISPEC`(v - x) cross (va - x:real^3)`NORM_POS_LE)
THEN REAL_ARITH_TAC];(*5*)

ABBREV_TAC`s1= min (s:real) (atn ((norm ((v - x) cross (va - x))) * inv((v - x) dot (va - x)))) / &2`
THEN ASSUME_TAC(ISPEC`(norm ((v - x) cross (u - x)) * inv ((v - x) dot (u - x))):real`ATN_BOUNDS)
THEN MP_TAC(REAL_ARITH`&0< s /\ s< pi/ &2/\ s1= min (s:real) (atn ((norm ((v - x) cross (va - x))) * inv((v - x) dot (va - x)))) / &2
 /\ &0< pi /\  &0< (atn ((norm ((v - x) cross (va - x))) * inv((v - x) dot (va - x))))
/\ (atn ((norm ((v - x) cross (va - x))) * inv((v - x) dot (va - x)))) < pi/ &2
==> &0<= s1 /\ s1< s /\  s <= pi/\ &0<s1 /\ s1<pi/ &2
/\  s1< (atn ((norm ((v - x) cross (va - x))) * inv((v - x) dot (va - x))))
`)
THEN REWRITE_TAC[PI_WORKS]
THEN RESA_TAC
THEN EXISTS_TAC`sin (s1) % (e1_fan x v va) + (cos s1) %(e3_fan x v va)+x :real^3 `
THEN REWRITE_TAC[dist;vector_norm;VECTOR_ARITH`(B+C+A)-A=(B+C:real^3)`; DOT_LADD;DOT_RADD;DOT_RMUL;DOT_LMUL;]
THEN MRESAL_TAC properties_coordinate[`x:real^3`;`v:real^3`;`va:real^3`][orthonormal]
THEN ONCE_REWRITE_TAC[DOT_SYM]
THEN ASSUME_TAC(ISPEC`s1:real`SIN_CIRCLE)
THEN ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`sin s * sin s + cos s * cos s =(sin s) pow 2 + (cos s) pow 2`;e3_fan;DOT_RMUL; ]
THEN ONCE_REWRITE_TAC[GSYM vector_norm;]
THEN REWRITE_TAC[DOT_SQUARE_NORM;]
THEN MRESAL1_TAC  SQRT_POW_2`&1`[REAL_ARITH`a pow 2 = a * a`;REAL_ARITH`&0<= &1`;]
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC SQRT_MUL[`&1`;`&1`][REAL_ARITH`&0<= &1`;]
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th); REAL_ARITH`&1* &1= &1`])
THEN RESA_TAC
THEN REDUCE_ARITH_TAC
THEN SUBGOAL_THEN`~(norm((v:real^3)-(x:real^3))= &0)` ASSUME_TAC
THENL(*6*)[ASM_REWRITE_TAC[NORM_EQ_0;VECTOR_ARITH`v-x=vec 0<=> x=v`];(*6*)
MP_TAC(ISPEC`norm((v:real^3)-(x:real^3))`REAL_MUL_LINV)
THEN REWRITE_TAC[REAL_ARITH`A*B*D *D=(B*D) *D *A`;REAL_ARITH`A>B<=> B<A`]
THEN RESA_TAC
THEN REDUCE_ARITH_TAC
THEN ASSUME_TAC(ISPEC`v-x:real^3`NORM_POS_LE)
THEN STRIP_TAC
THENL(*7*)[
MATCH_MP_TAC REAL_LT_LMUL
THEN MP_TAC(REAL_ARITH`~(norm (v - x:real^3) = &0)/\ &0 <= norm (v - x)==> &0< norm (v - x) `)
THEN RESA_TAC
THEN MATCH_MP_TAC COS_MONO_LT
THEN ASM_REWRITE_TAC[];
MRESA_TAC condition_to_in_aff_gt_by_angle[`x:real^3`;`v:real^3`;`va:real^3`;`s1:real`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[e3_fan]]]]]]]]);;

let scale_in_edges_fan=prove(`!(x:real^3) (v:real^3) (u:real^3) (w:real^3). 
DISJOINT {x} {v,u}
/\ w IN aff_gt {x} {v,u}
==> 
(?a t:real. &0<a /\ &0<t /\ t< &1
/\ a%(w-x) = (&1-t)% v+ t%u-x)`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[VECTOR_ARITH`w = t1 % x + t2 % v + t3 % u 
<=> w-x = ((t1+t2+t3)- &1) % x + (((t1+t2+t3) -t1)- t3)% (v-x) + t3 % (u-x):real^3`]
THEN ASM_REWRITE_TAC[REAL_ARITH`&1- &1= &0`] THEN REDUCE_VECTOR_TAC
THEN MP_TAC(REAL_ARITH`&0< t2 /\ &0< t3 /\ t1+t2+t3= &1 ==> ~(&1-t1= &0)/\ &0< &1- t1`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV `(&1-t1:real)`
THEN MRESA1_TAC REAL_LT_INV `(&1-t1:real)`
THEN MRESA_TAC REAL_LT_MUL [`inv(&1-t1:real)`;`t3:real`]
THEN MRESA_TAC REAL_LT_MUL [`inv(&1-t1:real)`;`t2:real`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[REAL_ARITH`inv (&1 - t1) * t2=inv (&1 - t1) * ((t1+t2+t3)-t1)- inv(&1-t1)*t3:real`]
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
w - x = (&1 - t1 - t3) % (v - x) + t3 % (u - x):real^3
==> (inv (&1- t1))%(w - x ) = (inv (&1-t1))%((&1 - t1 - t3) % (v - x) + t3 % (u - x))
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C= (A*B)%C`;VECTOR_ARITH`A%(B+C)=A%B+A%C`;REAL_ARITH`inv (&1 - t1) * (&1 - t1 - t3)=inv (&1 - t1) * (&1 - t1) - inv (&1 - t1) * (t3)`;VECTOR_ARITH`(&1-A)%(U-X)+A%(V-X)=(&1-A)%U+A%V-X`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`inv(&1- t1:real)`
THEN EXISTS_TAC`inv(&1-t1) *t3:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`A< &1<=> &0< &1- A`]);;


let aff_gt_imp_not_collinear=prove(`!x u v w:real^3.
~collinear{x,v,u}/\ w IN aff_gt{x,v} {u}==> ~collinear{x,v,w}`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(u:real^3) `;]
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC  AFF_GT_2_1[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN ASM_REWRITE_TAC[collinear_fan;aff;AFFINE_HULL_2;IN_ELIM_THM;]
THEN DISCH_THEN(LABEL_TAC"A")
THEN REPEAT STRIP_TAC
THEN REMOVE_THEN "A" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1 % x + t2 % v + t3 % u = u' % x + v' % v<=>
t3 % u = (u'-t1) % x + (v'-t2) % v`]
THEN MP_TAC(REAL_ARITH`&0<t3==> ~(t3= &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`t3:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`t3 % u = (u' - t1) % x + (v' - t2) % v:real^3
 ==> (inv (t3))%(t3 % u ) = (inv (t3))%( (u' - t1) % x + (v' - t2) % v)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`(inv (t3)) * (u' - t1):real`
THEN EXISTS_TAC`(inv (t3)) * (v' -t2):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (t3) * (u' - t1) + inv (t3) * (v'- t2)
= inv (t3) * (t3+ (u'+v')- (t1 + t2 +t3)):real`; REAL_ARITH`A+ &1- &1= A`]);;

let conditions_in_rcone_fan=prove(`!x v u w:real^3 s:real.
~collinear {x,v,u}/\ w IN aff_gt {x} {v,u} /\ &0<s /\ s< pi/ &2 /\u IN rcone_fan x v (cos s)==> w IN rcone_fan x v (cos s) `,

REWRITE_TAC[rcone_fan;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_TAC
THEN DISCH_TAC
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(u:real^3)`;]
THEN MRESAL_TAC AFF_GT_1_2[`(x:real^3)` ;` (v:real^3)`;` (u:real^3)  `;][IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[dist;VECTOR_ARITH`(t1 % x + t2 % v + t3 % u) - x=((t1+t2+t3)- &1) % x + t2 % (v-x) + t3 % (u-x)`;REAL_ARITH`&1 - &1= &0`;REAL_ARITH`A>B<=> B<A`;DOT_LADD;DOT_LMUL]
THEN REDUCE_VECTOR_TAC
THEN REDUCE_ARITH_TAC
THEN STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0< t3==> &0<= t3`) THEN RESA_TAC
THEN MRESA1_TAC REAL_ABS_REFL`t3:real`
THEN MRESA_TAC NORM_MUL[`t3:real`;`u-x:real^3`]
THEN MP_TAC(REAL_ARITH`&0< t2==> &0<= t2`) THEN RESA_TAC
THEN MRESA1_TAC REAL_ABS_REFL`t2:real`
THEN MRESA_TAC NORM_MUL[`t2:real`;`v-x:real^3`]
THEN MRESA_TAC REAL_LT_LMUL[`t3:real`;`norm (u - x) * norm (v - x:real^3) * (cos s):real`;`(u - x) dot (v - x:real^3)`]
THEN MRESA1_TAC COS_BOUNDS`s:real`
THEN MRESA1_TAC DOT_POS_LE`(v - x):real^3`
THEN MRESA_TAC REAL_LE_MUL[`t2:real`;`(v-x) dot (v-x:real^3)`;]
THEN MRESA_TAC REAL_LE_LMUL[`t2*((v - x:real^3) dot (v-x)):real`;`cos s:real`;`&1`]
THEN POP_ASSUM MP_TAC THEN REDUCE_ARITH_TAC 
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[DOT_SQUARE_NORM;]
THEN STRIP_TAC
THEN MP_TAC(REAL_ARITH` t3 * norm (u - x) * norm (v - x) * cos s< t3 * ((u - x) dot (v - x))
/\(t2 * norm (v - x) pow 2) * cos s <= t2 * ((v - x) dot (v - x))
==> (t2 * norm (v - x:real^3)  + t3 * norm (u - x)) * norm (v - x) * cos s< t2 * ((v - x) dot (v - x))+ t3 * ((u - x) dot (v - x))`)
THEN RESA_TAC
THEN MRESA_TAC NORM_TRIANGLE[`t2 %(v - x:real^3)`;`t3 % (u - x:real^3)`]
THEN MRESA1_TAC NORM_POS_LE`(v - x):real^3`
THEN MRESA1_TAC COS_POS_PI2`(s):real`
THEN MP_TAC(REAL_ARITH`&0< cos s:real ==> &0<= cos s`) THEN RESA_TAC
THEN MRESA_TAC REAL_LE_MUL[`norm  (v-x:real^3)`;`cos s:real`]
THEN MRESA_TAC REAL_LE_RMUL[`norm (t2 % (v - x) + t3 % (u - x):real^3):real`;`t2 * norm (v - x:real^3) + t3 * norm (u - x)`;`norm (v - x:real^3) * cos s`]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC);;

let exists_point_inside_domain_cone_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 s:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<s /\ s<pi/ &2
/\ fan80(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
==>
(?y:real^3. y IN rw_dart_fan x V E ((x:real^3),(u:real^3),(w:real^3),(w2:real^3)) (cos(s)) /\
azim x v u y< azim x v u w)`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "BE")
THEN DISCH_THEN(LABEL_TAC"EM")
THEN USE_THEN "BE" MP_TAC
THEN REWRITE_TAC[fan80]
THEN DISCH_TAC
THEN POP_ASSUM (fun th->   MRESA_TAC th [`u:real^3`;`w:real^3`]
THEN MRESA_TAC th [`v:real^3`;`u:real^3`])
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`]
THEN REMOVE_THEN "EM" (fun th-> MRESA1_TAC th `u:real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN DISJ_CASES_TAC(SET_RULE`set_of_edge u V E = {w} \/ ~(set_of_edge u V E = {w:real^3})`)
THENL(*1*)[
MRESA_TAC CARD_SING[`w:real^3`; `(set_of_edge u V E):real^3->bool`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th->REWRITE_TAC[SYM(th)])
THEN REMOVE_ASSUM_TAC THEN POP_ASSUM MP_TAC THEN ARITH_TAC;(*1*)

MRESA_TAC not_empty_rw_dart_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`u:real^3`; `w:real^3`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th `s:real`)
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[SET_RULE`~(A={})<=> ?y. y IN A`;rw_dart_fan;INTER; w_dart_fan;wedge;IN_ELIM_THM;]
THEN STRIP_TAC
THEN MP_TAC(REAL_ARITH`azim x u w y< azim x u w v==> azim x u w y<= azim x u w (v:real^3)`)
THEN RESA_TAC
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`u:real^3`;`w:real^3`;`y:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(LABEL_TAC"YEU EM")
THEN MP_TAC(REAL_ARITH`azim x u w y< azim x u w v/\ &0< azim x u w y/\ azim x u w v < pi /\ azim x u w v = azim x u w y + azim x u y v==>  ~(azim x u y (v:real^3)= &0)/\ &0< azim x u y v/\ azim x u y v < pi/\ ~(azim x u y (v:real^3)= pi)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`u:real^3`;`y:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,D,B}`]
THEN REPEAT DISCH_TAC
THEN MRESA_TAC NOT_COPLANAR_NOT_COLLINEAR[`x:real^3`;`v:real^3`;`y:real^3`;`u:real^3`]
THEN SUBGOAL_THEN`~(azim x v u (y:real^3)= &0)`ASSUME_TAC
THENL(*2*)[
STRIP_TAC
THEN MRESA_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`v:real^3`;`u:real^3`;`y:real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,D,B}`]
THEN MRESA_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`u:real^3`;`y:real^3`;`v:real^3`]
THEN POP_ASSUM MATCH_MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[];(*2*)

SUBGOAL_THEN`~(azim x v u (y:real^3)= pi)`ASSUME_TAC
THENL(*3*)[

STRIP_TAC
THEN MRESA_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`v:real^3`;`u:real^3`;`y:real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,D,B}`]
THEN MRESA_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`u:real^3`;`y:real^3`;`v:real^3`]
THEN POP_ASSUM MATCH_MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[];(*3*)

MP_TAC(REAL_ARITH`~(azim x v u y=pi)==>pi< azim x v u y  \/ azim x v u (y:real^3) < pi`)
THEN RESA_TAC
THENL(*4*)[

MRESA_TAC AZIM_COMPL[`x:real^3`;`v:real^3`;`u:real^3`;`y:real^3`]
THEN MRESA_TAC azim[`x:real^3`;`v:real^3`;`u:real^3`;`y:real^3`]
THEN MP_TAC(REAL_ARITH`pi< azim x v u y /\ azim x v u y < &2 * pi /\ azim x v y u = &2 * pi - azim x v u y
==> azim x v y u< pi/\ &0<azim x v y u  `)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN REPEAT STRIP_TAC
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`v:real^3`;`u:real^3`;`y:real^3`]
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`u:real^3`;`v:real^3`;`y:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`;CROSS_TRIPLE;]
THEN ONCE_REWRITE_TAC[CROSS_SKEW]
THEN ASM_REWRITE_TAC[DOT_LNEG]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;(*4*)
	
DISJ_CASES_TAC(REAL_ARITH`azim x v u w <= azim x v u (y:real^3) \/ azim x v u y < azim x v u w`)
THENL(*5*)[
ABBREV_TAC`a1=(v-x):real^3`
THEN ABBREV_TAC`a2=(y-x):real^3`
THEN ABBREV_TAC`a3=(u-x) :real^3`
THEN ABBREV_TAC`a4=w-x:real^3`
THEN ABBREV_TAC`va=a1 cross a4:real^3`
THEN ABBREV_TAC`vb=a2 cross a3:real^3`
THEN ABBREV_TAC`v3= (vb:real^3) cross (va:real^3)+(x:real^3)`
THEN ABBREV_TAC`v4= &1/ &2 % v3+ &1/ &2 % u:real^3`
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`u:real^3`;`v:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`;CROSS_TRIPLE;]
THEN ONCE_REWRITE_TAC[CROSS_SKEW; ]
THEN ASM_REWRITE_TAC[DOT_LNEG] 
THEN STRIP_TAC
THEN SUBGOAL_THEN `v3 IN aff_gt {x,u} {y:real^3}` ASSUME_TAC
THENL(*6*)[
MRESA_TAC th3[`(x:real^3)` ;` (u:real^3)`;`(y:real^3) `;]
THEN MRESAL_TAC  AFF_GT_2_1[`x:real^3`;`u:real^3`;`y:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&1-(va:real^3) dot (a2:real^3)+va dot (a3:real^3)`
THEN EXISTS_TAC`((va:real^3) dot (a2:real^3))`
THEN EXISTS_TAC`(--((va:real^3) dot (a3:real^3)))`
THEN ASM_REWRITE_TAC[REAL_ARITH`(&1 - va dot a3 + va dot a2) + (va dot a3) + --(va dot a2) = &1`;VECTOR_ARITH`(&1-A+B)%X+ (A)%U+ (--B) %V=A%(U-X)- B%(V-X)+X`]
THEN EXPAND_TAC"v3"
THEN EXPAND_TAC"vb" 
THEN ONCE_REWRITE_TAC[CROSS_SKEW; ]
THEN REWRITE_TAC[CROSS_LAGRANGE;VECTOR_ARITH`--(A-B)+C=B-A+C:real^3`];(*6*)
SUBGOAL_THEN `(v4:real^3) IN aff_gt {x,u} {y:real^3}` ASSUME_TAC
THENL(*7*)[
POP_ASSUM MP_TAC 
THEN MRESA_TAC th3[`(x:real^3)` ;` (u:real^3)`;`(y:real^3) `;]
THEN MRESAL_TAC  AFF_GT_2_1[`x:real^3`;`u:real^3`;`y:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`&1/ &2 * t1:real`
THEN EXISTS_TAC`&1/ &2 * t2+ &1/ &2:real`
THEN EXISTS_TAC`&1/ &2*t3:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`&1 / &2 * t1 + (&1 / &2 * t2 + &1 / &2) + &1 / &2 * t3 = &1/ &2*(t1+t2+t3)+ &1/ &2`; REAL_ARITH`&1/ &2 * &1 + &1/ &2= &1`;VECTOR_ARITH`(&1 / &2 * t1) % x + (&1 / &2 * t2 + &1 / &2) % u + (&1 / &2 * t3) % y= &1/ &2 %(t1 % x + t2 % u + t3 % y)+ &1/ &2 % u`]
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC;(*7*)
MRESA_TAC aff_gt_imp_not_collinear[`x:real^3`;`y:real^3`;`u:real^3`; `v4:real^3`]
THEN MRESA_TAC AZIM_EQ_ALT[`x:real^3`;`u:real^3`;`w:real^3`; `v4:real^3`;`y:real^3`]
THEN EXISTS_TAC`v4:real^3`
THEN ASM_REWRITE_TAC[]
THEN SUBGOAL_THEN `v3 IN aff_gt {x,v} {w:real^3}` ASSUME_TAC
THENL(*8*)[

MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(w:real^3) `;]
THEN MRESAL_TAC  AFF_GT_2_1[`x:real^3`;`v:real^3`;`w:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&1-(vb:real^3) dot (a4:real^3)+vb dot (a1:real^3)`
THEN EXISTS_TAC`((vb:real^3) dot (a4:real^3))`
THEN EXISTS_TAC`(--((vb:real^3) dot (a1:real^3)))`
THEN ASM_REWRITE_TAC[REAL_ARITH`(&1 - vb dot a4 + vb dot a1) + (vb dot a4) + --(vb dot a1) = &1`;VECTOR_ARITH`(&1-A+B)%X+ (A)%U+ (--B) %V=A%(U-X)- B%(V-X)+X`]
THEN EXPAND_TAC"v3"
THEN EXPAND_TAC"va" 
THEN REWRITE_TAC[CROSS_LAGRANGE;VECTOR_ARITH`--(A-B)+C=B-A+C:real^3`]
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`v:real^3`;`y:real^3`;`u:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`;CROSS_TRIPLE;]
THEN ONCE_REWRITE_TAC[CROSS_SKEW; ]
THEN ASM_REWRITE_TAC[DOT_LNEG] 
THEN DISCH_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN MRESA_TAC azim[`x:real^3`;`v:real^3`;`u:real^3`;`y:real^3`]
THEN FIND_ASSUM(MP_TAC)`~(azim x v u (y:real^3)= &0)`
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;(*8*)

REMOVE_THEN "YEU EM"(fun th-> ASM_REWRITE_TAC[SYM(th)])
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC aff_gt_imp_not_collinear[`x:real^3`;`w:real^3`;`v:real^3`; `v3:real^3`]
THEN MRESA_TAC AZIM_EQ_ALT[`x:real^3`;`v:real^3`;`u:real^3`; `v3:real^3`;`w:real^3`]
THEN SUBGOAL_THEN`~(coplanar{x,v,u,v3:real^3})`ASSUME_TAC
THENL(*9*)[
STRIP_TAC
THEN MRESA_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`v:real^3`;`u:real^3`;`v3:real^3`]
THEN MRESA_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`];(*9*)

SUBGOAL_THEN`~(azim x u v3 (v:real^3)= &0)`ASSUME_TAC
THENL(*10*)[
POP_ASSUM MP_TAC
THEN MATCH_MP_TAC MONO_NOT
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MRESA_TAC AZIM_EQ_0_PI_IMP_COPLANAR[`x:real^3`;`u:real^3`;`v3:real^3`;`v:real^3`]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,D,B}`]
THEN ASM_REWRITE_TAC[];(*10*)
MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`v3:real^3`]
THEN MRESA_TAC AZIM_EQ_ALT[`x:real^3`;`u:real^3`;`v:real^3`; `v3:real^3`;`y:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`;]
THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(LABEL_TAC"YEU EM")
THEN MRESA_TAC AZIM_COMPL[`x:real^3`;`u:real^3`;`y:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`;]
THEN RESA_TAC
THEN MRESA_TAC AZIM_COMPL[`x:real^3`;`u:real^3`;`v3:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`;]
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&2 * pi - azim x u y v = &2 * pi - azim x u v3 v==> azim x u v3 v= azim x u y v  `)
THEN RESA_TAC
THEN MRESAL_TAC inequality4_aim_in_convex_fan[`x:real^3`;`v:real^3`;`u:real^3`;`v3:real^3`;`&1/ &2`][REAL_ARITH`&1- &1/ &2= &1/ &2/\ &0< &1/ &2 /\ &1/ &2< &1`;VECTOR_ARITH`A+B=B+A:real^3`]
THEN MATCH_MP_TAC  conditions_in_rcone_fan
THEN EXISTS_TAC `y:real^3`
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC th3[`(x:real^3)` ;` (u:real^3)`;`(y:real^3) `;]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`u:real^3`;`y:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&1/ &2 *(&1-(va:real^3) dot (a2:real^3)+va dot (a3:real^3))`
THEN EXISTS_TAC`&1/ &2 *((va:real^3) dot (a2:real^3))+ &1/ &2`
THEN EXISTS_TAC`&1/ &2 *(--((va:real^3) dot (a3:real^3)))`
THEN ASM_REWRITE_TAC[REAL_ARITH`&1 / &2 * t1 + (&1 / &2 * t2 + &1 / &2) + &1 / &2 * t3 = &1/ &2*(t1+t2+t3)+ &1/ &2`; REAL_ARITH`&1/ &2 * &1 + &1/ &2= &1`;VECTOR_ARITH`(&1 / &2 * t1) % x + (&1 / &2 * t2 + &1 / &2) % u + (&1 / &2 * t3) % y= &1/ &2 %(t1 % x + t2 % u + t3 % y)+ &1/ &2 % u`;REAL_ARITH`(&1 - va dot a3 + va dot a2) + (va dot a3) + --(va dot a2) = &1`;VECTOR_ARITH`(&1-A+B)%X+ (A)%U+ (--B) %V=A%(U-X)- B%(V-X)+X`]
THEN EXPAND_TAC"v4"
THEN EXPAND_TAC"v3"
THEN EXPAND_TAC"vb" 
THEN ONCE_REWRITE_TAC[CROSS_SKEW; ]
THEN REWRITE_TAC[CROSS_LAGRANGE;VECTOR_ARITH`--(A-B)+C=B-A+C:real^3`]
THEN MP_TAC(REAL_ARITH`&0< --(va dot (a3:real^3))==> &0 < &1 / &2 * --(va dot a3)`)
THEN RESA_TAC
THEN MATCH_MP_TAC(REAL_ARITH`&0<= (va dot (a2:real^3))==> &0 < &1 / &2 * (va dot a2)+ &1/ &2`)
THEN MRESA_TAC cross_dot_fully_surrounded_ge_fan[`x:real^3`;`v:real^3`;`y:real^3`;`w:real^3`]
THEN POP_ASSUM MATCH_MP_TAC
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`;`y:real^3`]
THEN REWRITE_TAC[azim]
THEN MATCH_MP_TAC(REAL_ARITH`azim x v u y = azim x v u w + azim x v w y/\ &0<= azim x v u w /\ azim x v u y < pi ==>azim x v w y <= pi`)
THEN ASM_REWRITE_TAC[azim]](*10*)](*9*)](*8*)](*7*)](*6*);(*5*)
EXISTS_TAC`y:real^3`
THEN REMOVE_THEN "YEU EM" (fun th-> REWRITE_TAC[SYM(th)])	
THEN ASM_REWRITE_TAC[]]]]]]);;
let cut_in_angle_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 y:real^3.
 ~coplanar {x,v,u,w} /\ ~collinear {x,u,y}
/\ &0< azim x u w v /\ azim x u w v< pi
/\ azim x u w y< azim x u w v /\ &0< azim x u w y
==> let a1=(v-x):real^3 in
    let a2=w-x:real^3 in
    let a3=(y-x):real^3 in
    let a4=(u-x) :real^3 in
        let va=a1 cross a2:real^3 in
    let vb=a3 cross a4:real^3 in
    let v3= (vb:real^3) cross (va:real^3)+(x:real^3)
in v3 IN aff_gt {x} {v,w:real^3}`,

REPEAT STRIP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN ABBREV_TAC`a1=(v-x):real^3`
THEN ABBREV_TAC`a2=(y-x):real^3`
THEN ABBREV_TAC`a3=(u-x) :real^3`
THEN ABBREV_TAC`a4=w-x:real^3`
THEN ABBREV_TAC`va=a1 cross a4:real^3`
THEN ABBREV_TAC`vb=a2 cross a3:real^3`
THEN ABBREV_TAC`v3= (vb:real^3) cross (va:real^3)+(x:real^3)`
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(w:real^3) `;]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`w:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&1-(vb:real^3) dot (a4:real^3)+vb dot (a1:real^3)`
THEN EXISTS_TAC`((vb:real^3) dot (a4:real^3))`
THEN EXISTS_TAC`(--((vb:real^3) dot (a1:real^3)))`
THEN ASM_REWRITE_TAC[REAL_ARITH`(&1 - vb dot a4 + vb dot a1) + (vb dot a4) + --(vb dot a1) = &1`;VECTOR_ARITH`(&1-A+B)%X+ (A)%U+ (--B) %V=A%(U-X)- B%(V-X)+X`]
THEN EXPAND_TAC"v3"
THEN EXPAND_TAC"va" 
THEN REWRITE_TAC[CROSS_LAGRANGE;VECTOR_ARITH`--(A-B)+C=B-A+C:real^3`]
THEN MP_TAC(REAL_ARITH`azim x u w v< pi
/\ azim x u w y< azim x u w v==>azim x u w (y:real^3)< pi`)
THEN RESA_TAC
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`u:real^3`;`y:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[CROSS_SKEW; ]
THEN ASM_REWRITE_TAC[DOT_LNEG] 
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE;] THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN ASM_REWRITE_TAC[DOT_LNEG;REAL_ARITH`--(--A)=A`]
THEN DISCH_TAC
THEN MP_TAC(REAL_ARITH`azim x u w y< azim x u w v==> azim x u w y<= azim x u w (v:real^3)`)
THEN RESA_TAC
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`u:real^3`;`w:real^3`;`y:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`azim x u w y< azim x u w v/\ &0< azim x u w y/\ azim x u w v < pi /\ azim x u w v = azim x u w y + azim x u y v==>  ~(azim x u y (v:real^3)= &0)/\ &0< azim x u y v/\ azim x u y v < pi/\ ~(azim x u y (v:real^3)= pi)`)
THEN RESA_TAC
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`u:real^3`;`v:real^3`;`y:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ONCE_REWRITE_TAC[CROSS_SKEW; ]
THEN ASM_REWRITE_TAC[DOT_LNEG] );;
let aff_gt_1_2_scale_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 a:real.
&0< a /\ a % (u-x)= w-x /\ ~collinear {x,w,v}
==> aff_gt {x} {u,v} =aff_gt {x} {w,v}`,
REPEAT GEN_TAC
THEN GEOM_ORIGIN_TAC `x:real^3`
THEN REDUCE_VECTOR_TAC
THEN REPEAT STRIP_TAC 
THEN MP_TAC(REAL_ARITH`&0<a:real==> ~(a= &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV `a:real`
THEN MRESA1_TAC REAL_LT_INV`a:real`
THEN FIND_ASSUM(MP_TAC)`a %u=w:real^3`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
a%u=w:real^3
==> (inv (a))%(a%u) = (inv (a))%(w)`)
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C:real^3`]
THEN REWRITE_TAC[VECTOR_ARITH`&1 % u= inv a % w<=> inv a % w= u`]
THEN STRIP_TAC
THEN MRESA_TAC COLLINEAR_SPECIAL_SCALE[`a:real`;`u:real^3`;`v:real^3`]
THEN MRESA_TAC th3[`((vec 0):real^3)` ;` (u:real^3)`;`(v:real^3) `;]
THEN MRESA_TAC th3[`((vec 0):real^3)` ;` (w:real^3)`;`(v:real^3) `;]
THEN MRESAL_TAC  AFF_GT_1_2[`(vec 0):real^3`;`u:real^3`;`v:real^3`][IN_ELIM_THM;EXTENSION]
THEN MRESAL_TAC  AFF_GT_1_2[`(vec 0):real^3`;`w:real^3`;`v:real^3`][IN_ELIM_THM]
THEN REDUCE_VECTOR_TAC
THEN GEN_TAC THEN EQ_TAC
THENL[ STRIP_TAC
THEN EXISTS_TAC`&1- inv a * t2-t3:real`
THEN EXISTS_TAC `inv a * t2:real`
THEN EXISTS_TAC `t3:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`&1 - inv a * t2 - t3 + inv a * t2 + t3 = &1`;VECTOR_ARITH`(A*B)%C=B%(A%C)`]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[];
STRIP_TAC
THEN EXISTS_TAC`&1- a * t2-t3:real`
THEN EXISTS_TAC `a * t2:real`
THEN EXISTS_TAC `t3:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`&1 - a * t2 - t3 + a * t2 + t3 = &1`;VECTOR_ARITH`(A*B)%C=B%(A%C)`]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]]);;


let exists_cut_rcone_fan_with_edge_run_fan=prove(
`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 s:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<s /\ s<pi/ &2
/\ fan80(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
==>
(?t:real. &0< t /\ t< &1 /\
~(rw_dart_fan x V E ((x:real^3),(u:real^3),(w:real^3),(sigma_fan x V E u w:real^3)) (cos(s)) INTER aff_gt {x} {v, (&1-t)%u+t%w}={}))`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "BE")
THEN DISCH_THEN(LABEL_TAC"EM")
THEN USE_THEN "BE" MP_TAC
THEN REWRITE_TAC[fan80]
THEN DISCH_TAC
THEN POP_ASSUM (fun th->   MRESA_TAC th [`u:real^3`;`w:real^3`]
THEN MRESA_TAC th [`v:real^3`;`u:real^3`])
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`]
THEN REMOVE_THEN "EM" (fun th-> MRESA1_TAC th `u:real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN DISJ_CASES_TAC(SET_RULE`set_of_edge u V E = {w} \/ ~(set_of_edge u V E = {w:real^3})`)
THENL[
MRESA_TAC CARD_SING[`w:real^3`; `(set_of_edge u V E):real^3->bool`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th->REWRITE_TAC[SYM(th)])
THEN REMOVE_ASSUM_TAC THEN POP_ASSUM MP_TAC THEN ARITH_TAC;
MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN ASM_REWRITE_TAC[SET_RULE`~(A={})<=> ?y. y IN A`;rw_dart_fan;INTER; w_dart_fan;wedge;IN_ELIM_THM;]
THEN MRESA_TAC inequality3_aim_in_convex_fan
[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;` w:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"NHO EM")
THEN MRESA_TAC exists_point_inside_domain_cone_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`; `w:real^3`;`s:real`]
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC 
THEN ASM_REWRITE_TAC[rw_dart_fan;INTER; w_dart_fan;wedge;IN_ELIM_THM;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`azim x u w y< azim x u w v==> azim x u w y<= azim x u w (v:real^3)`)
THEN RESA_TAC
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`u:real^3`;`w:real^3`;`y:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(LABEL_TAC"YEU EM")
THEN MP_TAC(REAL_ARITH`azim x u w y< azim x u w v/\ &0< azim x u w y/\ azim x u w v < pi /\ azim x u w v = azim x u w y + azim x u y v==>  ~(azim x u y (v:real^3)= &0)/\ &0< azim x u y v/\ azim x u y v < pi/\ ~(azim x u y (v:real^3)= pi)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`u:real^3`;`y:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,D,B}`]
THEN REPEAT DISCH_TAC
THEN MRESA_TAC NOT_COPLANAR_NOT_COLLINEAR[`x:real^3`;`v:real^3`;`y:real^3`;`u:real^3`]
THEN SUBGOAL_THEN`~(azim x v u (y:real^3)= &0)`ASSUME_TAC
THENL[STRIP_TAC
THEN MRESA_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`v:real^3`;`u:real^3`;`y:real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,D,B}`]
THEN MRESA_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`u:real^3`;`y:real^3`;`v:real^3`]
THEN POP_ASSUM MATCH_MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[];
MP_TAC(REAL_ARITH`&0<= azim x v u (y:real^3) /\ ~(azim x v u y= &0)==> &0< azim x v u y`)
THEN ASM_REWRITE_TAC[azim] THEN STRIP_TAC
THEN MRESA_TAC angle_is_smallpi_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN ABBREV_TAC`a1=(v-x):real^3`
THEN ABBREV_TAC`a2=(y-x):real^3`
THEN ABBREV_TAC`a3=(u-x) :real^3`
THEN ABBREV_TAC`a4=w-x:real^3`
THEN ABBREV_TAC`va=a2 cross a1:real^3`
THEN ABBREV_TAC`vb=a4 cross a3:real^3`
THEN ABBREV_TAC`v3=(va:real^3) cross (vb:real^3)+(x:real^3)`
THEN MRESA_TAC cut_in_angle_fan[`x:real^3`;`w:real^3`;`v:real^3`;`u:real^3`;`y:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,D,B}`]
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN RESA_TAC
THEN POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`] THEN DISCH_TAC
THEN MRESA_TAC scale_in_edges_fan[`(x:real^3)`;`(u:real^3)`;`(w:real^3)`;`(v3:real^3)`]
THEN EXISTS_TAC`t:real`
THEN ASM_REWRITE_TAC[]
THEN EXISTS_TAC`y:real^3`
THEN REMOVE_THEN "YEU EM" (fun th-> REWRITE_TAC[SYM(th)])
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(REAL_ARITH`&0<t:real==> ~(t= &0)`) THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0<a:real==> ~(a= &0)`) THEN RESA_TAC
THEN MRESA_TAC continuous_coplanar_fan [`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `t:real`)
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`(&1 - t) % u + t % w:real^3`]
THEN MRESAL_TAC aff_gt_1_2_scale_fan[`x:real^3`;`v:real^3`;`v3:real^3`;`(&1 - t) % u + t % w:real^3`;`a:real`][VECTOR_ARITH`(&1 - t) % u + t % w - x = ((&1 - t) % u + t % w) - x`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN RESA_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN MRESAL_TAC COLLINEAR_SPECIAL_SCALE[`a:real`;`(v3 - x):real^3`;`v-x:real^3`][VECTOR_ARITH`(&1 - t) % u + t % w - x = ((&1 - t) % u + t % w) - x`]
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC"a1"
THEN ONCE_REWRITE_TAC[GSYM COLLINEAR_3]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,C,A}`]
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN STRIP_TAC
THEN MRESA_TAC th3[`(x:real^3)` ;` (v3:real^3)`;`(v:real^3) `;]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v3:real^3`;`v:real^3`][IN_ELIM_THM;]
THEN EXPAND_TAC"v3"
THEN EXPAND_TAC"va"
THEN ONCE_REWRITE_TAC[CROSS_SKEW]
THEN REWRITE_TAC[CROSS_LAGRANGE;]
THEN MP_TAC(REAL_ARITH`azim x u w y < azim x u w v /\ azim x u w v< pi==> azim x u w y< pi`)
THEN RESA_TAC
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`u:real^3`;`y:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[CROSS_SKEW]
THEN ASM_REWRITE_TAC[DOT_LNEG]
THEN STRIP_TAC
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`u:real^3`;`v:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ONCE_REWRITE_TAC[CROSS_SKEW]
THEN ASM_REWRITE_TAC[DOT_LNEG]
THEN STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0< --(vb dot (a1:real^3))==> ~(--(vb dot (a1:real^3))= &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV `(--(vb dot (a1:real^3)):real)`
THEN MRESA1_TAC REAL_LT_INV `(--(vb dot (a1:real^3)):real)`
THEN MRESA_TAC REAL_LT_MUL [`inv(--(vb dot (a1:real^3)):real)`;`--(vb dot (a2:real^3)):real`]
THEN EXISTS_TAC`&1- inv (--(vb dot (a1:real^3)))- inv (--(vb dot a1)) * --(vb dot a2)`
THEN EXISTS_TAC`inv (--(vb dot (a1:real^3)))`
THEN EXISTS_TAC`inv (--(vb dot a1)) * --(vb dot (a2:real^3))`
THEN ASM_REWRITE_TAC[REAL_ARITH`&1- T1 -T2 +T1 +T2 = &1`;VECTOR_ARITH`A%(--(B%X-C%Y)+Z)=(A*(--B))%X-(A*(--C))%Y+A%Z:real^3`]
THEN EXPAND_TAC"a1"
THEN EXPAND_TAC"a2"
THEN VECTOR_ARITH_TAC]]);;





let aff_gt_in_rw_dart_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 y:real^3 s:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<s /\ s<pi/ &2
/\ y IN rw_dart_fan x V E ((x:real^3),(u:real^3),(w:real^3),(v:real^3)) (cos(s)) 
/\ fan80(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
==> aff_gt {x} {u,y} SUBSET rw_dart_fan x V E ((x:real^3),(u:real^3),(w:real^3),(sigma_fan x V E u w:real^3)) (cos(s)) `,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "CON")
THEN DISCH_THEN(LABEL_TAC "BE")
THEN DISCH_THEN(LABEL_TAC"EM")
THEN USE_THEN "BE" MP_TAC
THEN REWRITE_TAC[fan80]
THEN DISCH_TAC
THEN POP_ASSUM (fun th->   MRESA_TAC th [`u:real^3`;`w:real^3`]
THEN MRESA_TAC th [`v:real^3`;`u:real^3`])
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`]
THEN REMOVE_THEN "EM" (fun th-> MRESA1_TAC th `u:real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN DISJ_CASES_TAC(SET_RULE`set_of_edge u V E = {w} \/ ~(set_of_edge u V E = {w:real^3})`)
THENL[MRESA_TAC CARD_SING[`w:real^3`; `(set_of_edge u V E):real^3->bool`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th->REWRITE_TAC[SYM(th)])
THEN REMOVE_ASSUM_TAC THEN POP_ASSUM MP_TAC THEN ARITH_TAC;
REMOVE_THEN "CON" MP_TAC
THEN ASM_REWRITE_TAC[rw_dart_fan;INTER; SUBSET; w_dart_fan;wedge;IN_ELIM_THM;]
THEN STRIP_TAC THEN GEN_TAC THEN STRIP_TAC
THEN MRESA_TAC aff_gt_inter_aff_gt [`(x:real^3)`;`(u:real^3)`;`(y:real^3)`]
THEN MP_TAC(SET_RULE`aff_gt {x} {u, y} = aff_gt {x, u} {y} INTER aff_gt {x, y} {u}
/\ x' IN aff_gt {x} {u, y:real^3} ==> x' IN aff_gt {x,u} {y}`) THEN RESA_TAC
THEN MRESA_TAC aff_gt_imp_not_collinear[`x:real^3`;`y:real^3`;`u:real^3`; `x':real^3`]
THEN MRESA_TAC AZIM_EQ_ALT[`x:real^3`;`u:real^3`;`w:real^3`;`x':real^3`;`y:real^3`;]
THEN MATCH_MP_TAC  conditions_in_rcone_fan
THEN EXISTS_TAC`y:real^3`
THEN ASM_REWRITE_TAC[]]);;
let in_aff_gt_1_2=prove(`!x:real^3 v:real^3 u:real^3 t:real.
DISJOINT {x} {v,u} /\ &0< t /\ t< &1==>  (&1-t)% v+ t% u IN aff_gt {x} {v,u}`,
REPEAT STRIP_TAC
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM;]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1 - t:real`
THEN EXISTS_TAC`t:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`&0< &1 - t<=> t< &1`;REAL_ARITH`&0 + &1 - t + t= &1`]
THEN VECTOR_ARITH_TAC);;

let exists_rw_dart_inter_aff_gt1_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 s:real.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ &0<s /\ s<pi/ &2
/\ fan80(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
==>
(?h:real. &0< h /\
(!t:real. &0< t /\ t< h==> 
~(rw_dart_fan x V E ((x:real^3),(u:real^3),(w:real^3),(sigma_fan x V E u w:real^3)) (cos(s)) INTER aff_gt {x} {v, (&1-t)%u+t%w}={})
))`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "BE")
THEN DISCH_THEN(LABEL_TAC"EM")
THEN USE_THEN "BE" MP_TAC
THEN REWRITE_TAC[fan80]
THEN DISCH_TAC
THEN POP_ASSUM (fun th->   MRESA_TAC th [`u:real^3`;`w:real^3`]
THEN MRESA_TAC th [`v:real^3`;`u:real^3`])
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`]
THEN REMOVE_THEN "EM" (fun th-> MRESA1_TAC th `u:real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN DISJ_CASES_TAC(SET_RULE`set_of_edge u V E = {w} \/ ~(set_of_edge u V E = {w:real^3})`)
THENL[ MRESA_TAC CARD_SING[`w:real^3`; `(set_of_edge u V E):real^3->bool`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th->REWRITE_TAC[SYM(th)])
THEN REMOVE_ASSUM_TAC THEN POP_ASSUM MP_TAC THEN ARITH_TAC;
MRESA_TAC exists_cut_rcone_fan_with_edge_run_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` v:real^3`;`u:real^3`;`w:real^3`;`s:real`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[SET_RULE`~(A={})<=> ?y. y IN A`;INTER;IN_ELIM_THM;]
THEN STRIP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"OK") THEN DISCH_TAC
THEN USE_THEN "OK" MP_TAC
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[rw_dart_fan;]
THEN ASM_REWRITE_TAC[INTER; w_dart_fan;wedge;IN_ELIM_THM;]
THEN STRIP_TAC
THEN EXISTS_TAC`t:real`
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN ABBREV_TAC`vt= (&1 - t':real) % u + t' % w :real^3`
THEN ABBREV_TAC`a1=(v-x):real^3`
THEN ABBREV_TAC`a2=vt-x:real^3`
THEN ABBREV_TAC`a3=(y-x):real^3`
THEN ABBREV_TAC`a4=(u-x) :real^3`
THEN ABBREV_TAC`va=a1 cross a2:real^3`
THEN ABBREV_TAC`vb=a3 cross a4:real^3`
THEN ABBREV_TAC`v3=(vb:real^3) cross (va:real^3)+(x:real^3)`
THEN MP_TAC(REAL_ARITH`&0< t'/\ t'< t /\ t< &1==> ~(t'= &0) /\ t'< &1`) THEN RESA_TAC
THEN MRESA_TAC in_aff_gt_1_2[`x:real^3`;`u:real^3`;`w:real^3 `;`t':real`]
THEN MRESA_TAC aff_gt_inter_aff_gt [`(x:real^3)`;`(u:real^3)`;`(w:real^3)`]
THEN MP_TAC(SET_RULE`aff_gt {x} {u, w} = aff_gt {x, u} {w} INTER aff_gt {x, w} {u}
/\ vt IN aff_gt {x} {u, w:real^3} ==> vt IN aff_gt {x,u} {w}`) THEN RESA_TAC
THEN MRESA_TAC aff_gt_imp_not_collinear[`x:real^3`;`w:real^3`;`u:real^3`; `vt:real^3`]
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`u:real^3`;`v:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,B,C}`]
THEN ASM_REWRITE_TAC[DE_MORGAN_THM] THEN STRIP_TAC
THEN MRESA_TAC AZIM_EQ_ALT[`x:real^3`;`u:real^3`;`v:real^3`;`vt:real^3`;`w:real^3`;]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MRESA_TAC AZIM_COMPL[`x:real^3`;`u:real^3`;`v:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`;]
THEN RESA_TAC
THEN MRESA_TAC AZIM_COMPL[`x:real^3`;`u:real^3`;`v:real^3`;`vt:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`;]
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN STRIP_TAC
THEN MRESA_TAC AZIM_EQ_ALT[`x:real^3`;`u:real^3`;`y:real^3`;`vt:real^3`;`w:real^3`;]
THEN MP_TAC(REAL_ARITH`&0< azim x u w y /\ azim x u w y< azim x u w v /\ azim x u w v< pi==>
azim x u w y<= azim x u w v/\ ~(azim x u w y = &0 \/ azim x u w y= pi)`) THEN RESA_TAC
THEN MRESA_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`u:real^3`;`w:real^3`;`y:real^3`]
THEN MRESA_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`u:real^3`;`y:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,B,D,C}`]
THEN ASM_REWRITE_TAC[DE_MORGAN_THM] THEN STRIP_TAC
THEN MRESA_TAC AZIM_COMPL[`x:real^3`;`u:real^3`;`y:real^3`;`w:real^3`]
THEN MRESA_TAC AZIM_COMPL[`x:real^3`;`u:real^3`;`y:real^3`;`vt:real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN STRIP_TAC
THEN MRESA_TAC continuous_coplanar_fan [`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `t':real`)
THEN MRESA_TAC cut_in_angle_fan[`x:real^3`;`v:real^3`;`u:real^3`;`vt:real^3`;`y:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,D,B}`]
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN RESA_TAC
THEN MRESA_TAC aff_gt_in_rw_dart_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`y:real^3`;`s:real`]
THEN EXISTS_TAC`v3:real^3`
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC(SET_RULE`aff_gt {x} {u, y} SUBSET rw_dart_fan x V E (x,u,w,v) (cos s)
/\ v3 IN aff_gt {x} {u, y} ==> v3 IN rw_dart_fan x V E (x,u,w,v) (cos s)`)
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC cut_in_angle_fan[`x:real^3`;`y:real^3`;`v:real^3`;`u:real^3`;`vt:real^3`]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[CROSS_SKEW]
THEN ASM_REWRITE_TAC[CROSS_LNEG;CROSS_RNEG]
THEN ONCE_REWRITE_TAC[GSYM CROSS_RNEG]
THEN ONCE_REWRITE_TAC[GSYM CROSS_SKEW]
THEN RESA_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN POP_ASSUM  MATCH_MP_TAC 
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`u:real^3`;`w:real^3`;`y:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(LABEL_TAC"NHO EM")
THEN MP_TAC(REAL_ARITH`azim x u w y< azim x u w v/\ &0< azim x u w y/\ azim x u w v < pi /\ azim x u w v = azim x u w y + azim x u y v==>  ~(azim x u y (v:real^3)= &0 \/ azim x u y (v:real^3)= pi)`)
THEN RESA_TAC
THEN MRESA_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`u:real^3`;`y:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN RESA_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,C,B}`]
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`u:real^3`;`v:real^3`;`y:real^3`]
THEN MRESA_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`v:real^3`;`u:real^3`;`y:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`]
THEN ASM_REWRITE_TAC[DE_MORGAN_THM] THEN STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0<=azim x v u y /\ ~(azim x v u y= &0)==> &0< azim x v u y`)
THEN ASM_REWRITE_TAC[azim] THEN RESA_TAC
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`vt:real^3`]
THEN MRESAL_TAC AZIM_EQ_0_PI_EQ_COPLANAR[`x:real^3`;`v:real^3`;`u:real^3`;`vt:real^3`][DE_MORGAN_THM]
THEN MP_TAC(REAL_ARITH`&0<=azim x v u vt /\ ~(azim x v u vt= &0)==> &0< azim x v u vt`)
THEN ASM_REWRITE_TAC[azim] THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0<t==> ~(t= &0)`) THEN RESA_TAC
THEN MRESA_TAC continuous_coplanar_fan [`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `t:real`)
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`(&1-t) %u+ t%w:real^3`]
THEN MRESA_TAC aff_gt_inter_aff_gt [`(x:real^3)`;`(v:real^3)`;`((&1-t) %u+ t%w:real^3)`]
THEN MP_TAC(SET_RULE`aff_gt {x} {v, (&1-t) %u+ t%w} = aff_gt {x, v} {(&1-t) %u+ t%w} INTER aff_gt {x, (&1-t) %u+ t%w} {v}
/\ y IN aff_gt {x} {v, (&1-t) %u+ t%w:real^3} ==> y IN aff_gt {x,v} {(&1-t) %u+ t%w}`) THEN RESA_TAC
THEN MRESA_TAC AZIM_EQ_ALT[`x:real^3`;`v:real^3`;`u:real^3`;`y:real^3`;`(&1-t) %u+ t%w:real^3`;]
THEN MRESA_TAC inequality4_aim_in_convex_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`;`t:real`]
THEN MRESA_TAC angle_is_smallpi_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MP_TAC(REAL_ARITH`azim x v u w< pi /\ azim x v u ((&1 - t) % u + t % w) < azim x v u w
==> azim x v u ((&1 - t) % u + t % w) < pi`) THEN RESA_TAC
THEN ABBREV_TAC`vs= (&1 - t:real) % u + t % w :real^3`
THEN MRESA_TAC in_aff_gt_1_2[`x:real^3`;`u:real^3`;`w:real^3 `;`t:real`]
THEN MP_TAC(SET_RULE`aff_gt {x} {u, w} = aff_gt {x, u} {w} INTER aff_gt {x, w} {u}
/\ vs IN aff_gt {x} {u, w:real^3} ==> vs IN aff_gt {x,u} {w}`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`t:real`
THEN MRESA1_TAC REAL_LT_INV`t:real`
THEN MRESA_TAC REAL_LT_MUL[`inv t:real`;`t':real`]
THEN MRESA_TAC REAL_LT_LMUL[`inv t:real`;`t':real`;`t:real`]

THEN MRESA_TAC AZIM_EQ_ALT[`x:real^3`;`u:real^3`;`v:real^3`;`vs:real^3`;`w:real^3`;]
THEN REMOVE_THEN"NHO EM" MP_TAC
THEN DISCH_TAC THEN REMOVE_ASSUM_TAC
THEN MRESA_TAC AZIM_COMPL[`x:real^3`;`u:real^3`;`v:real^3`;`vs:real^3`]
THEN MRESA_TAC AZIM_COMPL[`x:real^3`;`u:real^3`;`v:real^3`;`w:real^3`]
THEN MRESA_TAC inequality4_aim_in_convex_fan[`x:real^3`;`v:real^3`;`u:real^3`;`vs:real^3`;`inv (t) * t':real`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM(th)])
THEN STRIP_TAC
THEN MATCH_MP_TAC(SET_RULE`azim x v u ((&1 - inv t * t') % u + (inv t * t') % vs) < azim x v u vs /\
(&1 - inv t * t') % u + (inv t * t') % vs = vt ==> azim x v u vt < azim x v u vs`)
THEN ASM_REWRITE_TAC[]
THEN EXPAND_TAC"vs"
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(&1 - inv t * t') % u + (inv t * t') % ((&1 - t) % u + t % w)
= (&1 - (inv t * t) * t') % u + ((inv t * t) *t') % w`;REAL_ARITH`&1* A=A`]]);;


let there_exists_component_contain_aff_gt_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ fan80(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
==>
 (?h:real. &0< h /\ (?y:real^3.
 aff_gt {x} {v, (&1-h)%u+h%w} SUBSET connected_component (yfan(x,V,E)) y ))`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "BE")
THEN DISCH_THEN(LABEL_TAC"EM")
THEN USE_THEN "BE" MP_TAC
THEN REWRITE_TAC[fan80]
THEN DISCH_TAC
THEN POP_ASSUM (fun th->   MRESA_TAC th [`u:real^3`;`w:real^3`]
THEN MRESA_TAC th [`v:real^3`;`u:real^3`])
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`]
THEN REMOVE_THEN "EM" (fun th-> MRESA1_TAC th `u:real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN MRESA_TAC fan_run_in_small_is_subset_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th `h/ &2`)
THEN POP_ASSUM MP_TAC
THEN MP_TAC(REAL_ARITH`&0< h==> ~(h/ &2 = &0)/\ &0< h/ &2 /\ h/ &2 < h`) THEN RESA_TAC
THEN DISCH_TAC
THEN EXISTS_TAC`h/ &2 :real`
THEN ASM_REWRITE_TAC[]
THEN EXISTS_TAC`&1/ &2 % v + &1/ &2 %((&1 - h / &2) % u + h / &2 % w:real^3)`
THEN MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL 
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC CONVEX_AFF_GT[`{x:real^3}`;`{v, (&1 - h / &2) % u + h / &2 % w:real^3}`]
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC continuous_coplanar_fan [`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `h/ &2:real`)
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`(&1-h/ &2) %u+  h / &2 %w:real^3`]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(&1 - h/ &2) % (u:real^3) + h / &2 % (w:real^3)`;]
THEN MRESAL_TAC in_aff_gt_1_2[`x:real^3`;`v:real^3`;`(&1 - h / &2) % u + h / &2 % w:real^3 `;`&1/ &2:real`]
[REAL_ARITH`&1/ &2 < &1 /\ &0< &1/ &2 `; REAL_ARITH`&1 - &1/ &2 = &1/ &2` ]
THEN MATCH_MP_TAC CONVEX_CONNECTED 
THEN ASM_REWRITE_TAC[]);;


let CONNECTED_COMPONENT_OF_SUBSET = prove
 (`!s t x y. s SUBSET t /\ connected_component s x y
           ==> connected_component t x y`,
  REWRITE_TAC[connected_component] THEN SET_TAC[]);;
let connected_component_of_faces_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
==>
dart_leads_into x V E v u = dart_leads_into x V E u w`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "BE")
THEN DISCH_THEN(LABEL_TAC"con")
THEN USE_THEN "con" MP_TAC
THEN REWRITE_TAC[fan80]
THEN DISCH_TAC
THEN POP_ASSUM (fun th->   MRESA_TAC th [`u:real^3`;`w:real^3`]
THEN MRESA_TAC th [`v:real^3`;`u:real^3`])
THEN MRESA_TAC exists_leads_into_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;` u:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"ANH")
THEN MRESA_TAC exists_leads_into_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`u:real^3`;` w:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"YEU")
THEN ABBREV_TAC`h00= min h' (pi/ &2) / &2 :real`
THEN MP_TAC(REAL_ARITH`&0< h' /\ &0< pi /\ h00= min h' (pi/ &2) / &2 :real==> &0< h00 /\ h00 < pi / &2`)
THEN REWRITE_TAC[PI_WORKS]
THEN RESA_TAC
THEN MRESA_TAC exists_rw_dart_inter_aff_gt1_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` v:real^3`;` u:real^3`;` w:real^3`;`h00:real`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[SET_RULE`~(A={})<=> ?y. y IN A`]
THEN DISCH_THEN (LABEL_TAC"EM")
THEN MRESA_TAC exists_rw_dart_inter_aff_gt_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` v:real^3`;` u:real^3`;` w:real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[SET_RULE`~(A={})<=> ?y. y IN A`]
THEN DISCH_THEN (LABEL_TAC"NHIEU")
THEN MRESA_TAC fan_run_in_small_is_subset_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"HA")
THEN ABBREV_TAC`h1' = min h'' h''' / &2 :real`
THEN ABBREV_TAC`h1= min h1' h'''' / &2 :real`
THEN MP_TAC(REAL_ARITH`&0< h'' /\ &0 < h''' /\ h1'= min h'' h''' / &2 ==> &0< h1' /\ h1'< h'' /\ h1' < h'''`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0< h1'/\ h1'< h'' /\ h1' < h''' /\ &0 < h'''' /\ h1= min h1' h'''' / &2 ==> &0< h1 /\ h1< h'' /\ h1 < h''' /\ h1< h''''`)
THEN RESA_TAC
THEN ABBREV_TAC`h2= min h (pi/ &2) / &2 :real`
THEN MP_TAC(REAL_ARITH`&0< h /\ &0< pi /\ h2= min h (pi/ &2) / &2 :real==> &0< h2 /\ h2 < pi / &2`)
THEN REWRITE_TAC[PI_WORKS]
THEN RESA_TAC
THEN REMOVE_THEN "NHIEU" (fun th-> MRESAL1_TAC th `h1:real`[INTER;IN_ELIM_THM])
THEN POP_ASSUM (fun th-> MRESA1_TAC th `h2:real`)
THEN MP_TAC(REAL_ARITH`&0< h2 /\ h2= min h (pi/ &2) / &2 :real==>  h2 < h`)
THEN RESA_TAC
THEN USE_THEN "ANH" (fun th -> MP_TAC(ISPECL[`h2:real`;`y:real^3`]th))
THEN POP_ASSUM (fun th-> REWRITE_TAC[th])
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"LAM")
THEN REMOVE_THEN "EM" (fun th-> MRESAL1_TAC th `h1:real`[INTER;IN_ELIM_THM])
THEN MP_TAC(REAL_ARITH`&0< h00 /\ h00= min h' (pi/ &2) / &2 :real==>  h00 < h'`)
THEN RESA_TAC
THEN USE_THEN "YEU" (fun th -> MP_TAC(ISPECL[`h00:real`;`y':real^3`]th))
THEN POP_ASSUM (fun th-> REWRITE_TAC[th])
THEN RESA_TAC
THEN SUBGOAL_THEN `U=U':real^3->bool` ASSUME_TAC
THENL[
REMOVE_THEN "LAM" (fun th-> REWRITE_TAC[SYM(th)])
THEN POP_ASSUM  (fun th-> REWRITE_TAC[SYM(th)])
THEN REWRITE_TAC[CONNECTED_COMPONENT_EQ_EQ]
THEN MRESA_TAC CONVEX_AFF_GT[`{x:real^3}`;`{v, (&1 - h1) % u + h1  % w:real^3}`]
THEN MRESA_TAC CONVEX_CONNECTED[`aff_gt {x} {v, (&1 - h1) % u + h1 % w}:real^3->bool`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[CONNECTED_IFF_CONNECTED_COMPONENT]
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th [`y:real^3`;`y':real^3`])
THEN REMOVE_THEN "HA" (fun th-> MRESA1_TAC th `h1:real`)
THEN MRESA_TAC CONNECTED_COMPONENT_OF_SUBSET[`aff_gt {x} {v, (&1 - h1) % u + h1 % w:real^3}:real^3-> bool`;`yfan (x:real^3,(V:real^3->bool),(E:(real^3->bool)->bool)):real^3->bool`;`y:real^3`;`y':real^3`]
THEN ASM_TAC THEN SET_TAC[];
SUBGOAL_THEN`dart_leads_into x V E v u= U:real^3->bool` ASSUME_TAC
THENL[
REMOVE_ASSUM_TAC
THEN MATCH_MP_TAC unique_dart_leads_into
THEN ASM_REWRITE_TAC[]
THEN EXISTS_TAC`h:real`
THEN ASM_REWRITE_TAC[];

SUBGOAL_THEN`dart_leads_into x V E u w= U':real^3->bool` ASSUME_TAC
THENL[ MATCH_MP_TAC unique_dart_leads_into
THEN ASM_REWRITE_TAC[]
THEN EXISTS_TAC`h':real`
THEN ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[]]]]);;



(************************)







let dart_leads_into1 = new_definition 
    `dart_leads_into1 (x,V,E) (v,u) = @s.  s IN topological_component_yfan (x,V,E) /\
	(?eps. (eps < &1) /\ 
	   rw_dart_fan x V E (x,v,u,sigma_fan x V E v u) eps  SUBSET s)`;;

let dartset_leads_into = new_definition
  `dartset_leads_into (x,V,E) ds = 
    @s. (!y. (y IN ds) ==> (s = dart_leads_into1 (x,V,E) y))`;;


let exists_dartset_leads_into_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
==>
?s:real^3->bool. !y. y IN ds==> s= dart_leads_into x V E (pr2 y) (pr3 y)`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC hypermap_of_fan_rep[`x:real^3`;`V:real^3->bool`;`(E:(real^3->bool)->bool)`;`(\t. res (t x V E) (d1_fan (x,V,E)))`]
THEN ASM_REWRITE_TAC[face_set;set_of_orbits;IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`dart_leads_into x V E (pr2 x') (pr3 x'):real^3->bool`
THEN ASM_REWRITE_TAC[orbit_map;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`~(x' IN d1_fan (x,V,E) )\/ (x' IN d1_fan (x:real^3,V,E))`)
THENL[MRESA_TAC id_power_enf_fan[`x:real^3`;`V:real^3->bool`;`(E:(real^3->bool)->bool) `;`x':real^3#real^3#real^3#real^3` ;`n:num`;`(\t. res (t x V E) (d1_fan (x,V,E)))`]
THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[];
SPEC_TAC (`y:real^3#real^3#real^3#real^3`,`y:real^3#real^3#real^3#real^3`)
THEN SPEC_TAC (`n:num`,`n:num`)
THEN INDUCT_TAC
THENL[
REWRITE_TAC[POWER_0;I_THM]
THEN SET_TAC[];

POP_ASSUM MP_TAC
THEN MRESA_TAC into_domain_power_efn_fan[`x:real^3`;`V:real^3->bool`;`(E:(real^3->bool)->bool) `;`n:num`;`(\t. res (t x V E) (d1_fan (x,V,E)))`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th`x':real^3#real^3#real^3#real^3` )
THEN MRESA_TAC into_domain_power_efn_fan[`x:real^3`;`V:real^3->bool`;`(E:(real^3->bool)->bool) `;`SUC n:num`;`(\t. res (t x V E) (d1_fan (x,V,E)))`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th`x':real^3#real^3#real^3#real^3` )
THEN ABBREV_TAC`y1=(f1_fan x V E POWER n) (x':real^3#real^3#real^3#real^3)`
THEN DISCH_THEN(LABEL_TAC "EM")
THEN ASM_REWRITE_TAC[COM_POWER;o_DEF]
THEN POP_ASSUM (fun th-> MRESAL1_TAC th`y1:real^3#real^3#real^3#real^3`[ARITH_RULE`(n:num)>= 0`])
THEN REPEAT STRIP_TAC
THEN MRESA_TAC into_domain1_power_efn_fan[`x:real^3`;`V:real^3->bool`;`(E:(real^3->bool)->bool)`;`n:num`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th`x':real^3#real^3#real^3#real^3`)
THEN MRESA_TAC properties_of_f1_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`y:real^3#real^3#real^3#real^3`; `y1:real^3#real^3#real^3#real^3`]
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"EM") THEN REPEAT STRIP_TAC
THEN ABBREV_TAC`u=pr2 (f1_fan x V E y1):real^3`
THEN ABBREV_TAC`w=pr3 (f1_fan x V E y1):real^3`
THEN ABBREV_TAC`v=sigma_fan x V E u w:real^3`
THEN REMOVE_THEN "EM" MP_TAC THEN ASM_REWRITE_TAC[] THEN STRIP_TAC
THEN MRESA_TAC connected_component_of_faces_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`; `w:real^3`]]]);;



let dartset_leads_into_fan = new_definition
  `dartset_leads_into_fan x V E ds = 
    @s. (!y. (y IN ds) ==> (s = dart_leads_into x V E (pr2 y) (pr3 y)))`;;


let DARTSET_LEADS_INTO_FAN=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
==>
(!y. y IN ds==> dartset_leads_into_fan x V E ds= dart_leads_into x V E (pr2 y) (pr3 y))`,
REPEAT GEN_TAC THEN STRIP_TAC 
THEN ONCE_REWRITE_TAC[dartset_leads_into_fan]
THEN MRESA_TAC exists_dartset_leads_into_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;
`ds:real^3#real^3#real^3#real^3->bool`]
THEN SELECT_ELIM_TAC 
THEN EXISTS_TAC`s:real^3->bool`
 THEN ASM_REWRITE_TAC[]);;







let UNIQUE_DARTSET_LEADS_INTO_FAN=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds s.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ (!y. y IN ds==> s= dart_leads_into x V E (pr2 y) (pr3 y))
==> dartset_leads_into_fan x V E ds= s`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC FACE_FAN_NOT_EMPTY[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;
`ds:real^3#real^3#real^3#real^3->bool`]
THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[SET_RULE`~(A={})<=> ?y. y IN A`]
THEN STRIP_TAC
THEN MRESA_TAC DARTSET_LEADS_INTO_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;
`ds:real^3#real^3#real^3#real^3->bool`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `y:real^3#real^3#real^3#real^3`)
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> MRESA1_TAC th `y:real^3#real^3#real^3#real^3`));;



let equality_dart_leads_into=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds y y1.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ y IN ds /\ y1 IN ds

==>dart_leads_into x V E (pr2 y) (pr3 y)= dart_leads_into x V E (pr2 y1) (pr3 y1)`,
REPEAT STRIP_TAC
THEN MRESA_TAC exists_dartset_leads_into_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;
`ds:real^3#real^3#real^3#real^3->bool`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `y:real^3#real^3#real^3#real^3` THEN ASSUME_TAC(th))
THEN POP_ASSUM (fun th-> MRESA1_TAC th `y1:real^3#real^3#real^3#real^3`));;

let UNIQUE_DARTSET_LEADS_INTO1_FAN=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds s y.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ y IN ds /\ s= dart_leads_into x V E (pr2 y) (pr3 y)
==> dartset_leads_into_fan x V E ds= s`,
REPEAT STRIP_TAC
THEN SUBGOAL_THEN`(!y. y IN ds==> s= dart_leads_into (x:real^3) V E (pr2 y) (pr3 y))` ASSUME_TAC
THENL[REPEAT STRIP_TAC
THEN MRESA_TAC equality_dart_leads_into[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;
`ds:real^3#real^3#real^3#real^3->bool`;`y:real^3#real^3#real^3#real^3`;`y':real^3#real^3#real^3#real^3`];
MRESA_TAC UNIQUE_DARTSET_LEADS_INTO_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;
`ds:real^3#real^3#real^3#real^3->bool`;`s:real^3->bool`]]);;

let exists_point_dart_leads_into_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
==> ?y. y IN ds /\ dartset_leads_into_fan x V E ds =dart_leads_into x V E (pr2 y) (pr3 y)`,

REPEAT STRIP_TAC
THEN MRESA_TAC FACE_FAN_NOT_EMPTY[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;
`ds:real^3#real^3#real^3#real^3->bool`]
THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[SET_RULE`~(A={})<=> ?y. y IN A`]
THEN STRIP_TAC
THEN EXISTS_TAC`y:real^3#real^3#real^3#real^3`
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC UNIQUE_DARTSET_LEADS_INTO1_FAN [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;
`ds:real^3#real^3#real^3#real^3->bool`;`(dart_leads_into x V E (pr2 y) (pr3 y)):real^3->bool`;`y:real^3#real^3#real^3#real^3`]);;





let dartset_leads_into_is_topological_component_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
==> dartset_leads_into_fan x V E ds IN topological_component_yfan (x,V,E)`,
REPEAT STRIP_TAC
THEN MRESA_TAC exists_point_dart_leads_into_fan [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;
`ds:real^3#real^3#real^3#real^3->bool`;]
THEN MRESA_TAC properties_of_elements_in_face_fully_surroundedfan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`; `y:real^3#real^3#real^3#real^3`]
THEN MRESA_TAC dart_leads_into_fan_in_topological_component_yfan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`pr2(y):real^3`; `pr3(y):real^3`]);;

let dartset_leads_into_subset_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds:real^3#real^3#real^3#real^3->bool.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
==>
dartset_leads_into_fan x V E ds SUBSET yfan (x,V,E)`,
REPEAT STRIP_TAC
THEN MRESA_TAC dartset_leads_into_is_topological_component_yfan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[topological_component_yfan;IN_ELIM_THM;]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_SUBSET]);;



let RWXUYZZ=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds:real^3#real^3#real^3#real^3->bool.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
==>
(?s:real^3->bool. !y. y IN ds==> s= dart_leads_into x V E (pr2 y) (pr3 y))
/\ dartset_leads_into_fan x V E ds IN topological_component_yfan (x,V,E)`,
MESON_TAC[dartset_leads_into_is_topological_component_yfan;exists_dartset_leads_into_fan]);;


(************DWWUTKW*********)
let add_edge_graph_fan=prove(`!(V:A->bool) (E:(A->bool)->bool) v:A u:A.
 v IN V /\ u IN V /\ E1=E UNION {{v,u}} /\ UNIONS E SUBSET V==> UNIONS E1 SUBSET V`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[UNIONS; SUBSET; UNION; IN_ELIM_THM;IN_SING]
THEN DISCH_THEN(LABEL_TAC "YEU")
THEN REPEAT STRIP_TAC
THENL[SUBGOAL_THEN`(?u. u IN E /\ x IN (u:A->bool))`ASSUME_TAC
THENL[
EXISTS_TAC`u':A->bool` THEN ASM_REWRITE_TAC[];
REMOVE_THEN "YEU" (fun th-> MRESA1_TAC th `x:A`)];
ASM_TAC THEN SET_TAC[]]);;

let garph_add_edge_is_garph=prove(`!(V:A->bool) (E:(A->bool)->bool) v:A u:A.
 E1=E UNION {{v,u}} /\ ~(v=u) /\ graph E==> graph E1`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[GRAPH; UNION;IN_ELIM_THM;IN_SING]
THEN DISCH_THEN(LABEL_TAC"YEU")
THEN GEN_TAC THEN STRIP_TAC
THENL[REMOVE_THEN "YEU" (fun th-> MRESA1_TAC th `e:A->bool`);
ASM_REWRITE_TAC[HAS_SIZE] THEN STRIP_TAC
THEN MRESA_TAC CARD_2_FAN[`v:A`;`u:A`]
THEN SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY;
                 IN_INSERT; NOT_IN_EMPTY]]);;
let add_edge_into_collinear_fan=prove(`!x:real^3 (E:(real^3->bool)->bool) v:real^3 u:real^3.
~collinear {x,v,u} /\
(!e. e IN E ==> ~collinear ({x} UNION e))
==>
(!e. e IN E UNION {{v, u}} ==> ~collinear ({x} UNION e))`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV)[UNION;]
THEN REWRITE_TAC[IN_ELIM_THM;IN_SING]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN (LABEL_TAC"YEU")
THEN STRIP_TAC
THENL[REMOVE_THEN "YEU"(fun th-> MRESA1_TAC th `e:real^3->bool`);
ASM_REWRITE_TAC[SET_RULE`{X} UNION {A,B}={X,A,B}`]]);;

let condition_not_edge_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 ds.
FAN(x,V,E) /\ v IN V /\ u IN V /\ ~collinear {x,v,u}
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ e1 IN E /\ e2 = {v, u}
/\ aff_gt {x} {v,u} SUBSET dartset_leads_into_fan x V E ds
==> ~(e1=e2)`,


REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"YEU")
THEN STRIP_TAC
THEN MRESA_TAC aff_ge_eq_aff_gt_union_aff_ge[`(x:real^3)`;`(v:real^3)`;` (u:real^3)`]
THEN MRESA_TAC dartset_leads_into_subset_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` ds:real^3#real^3#real^3#real^3->bool`]
THEN MP_TAC(SET_RULE`dartset_leads_into_fan x V E ds SUBSET yfan (x,V,E)/\ aff_gt {x} {v, u} SUBSET dartset_leads_into_fan x V E ds==> aff_gt {x} {v, u} SUBSET yfan (x:real^3,V,E)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[yfan; SET_RULE`A SUBSET (:real^3) DIFF B <=> A INTER B={}`;]
THEN DISCH_TAC
THEN SUBGOAL_THEN`aff_ge {x} e1 SUBSET xfan(x:real^3,V:real^3->bool,E)` ASSUME_TAC
THENL[REWRITE_TAC[xfan;SUBSET;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC `e1:real^3->bool`
THEN ASM_REWRITE_TAC[]
THEN FIND_ASSUM MP_TAC(`(e1:real^3->bool) IN E`)
THEN REWRITE_TAC[IN]
THEN ASM_REWRITE_TAC[];
ASSUME_TAC(SET_RULE`aff_gt {x} {v, u} INTER
      (aff_gt {x} {v, u} UNION aff_ge {x} {v} UNION aff_ge {x} {u})
=aff_gt {x} {v, u} `)
THEN MP_TAC(SET_RULE`aff_gt {x:real^3} {v, u} INTER xfan (x,V:real^3->bool,E) = {}/\aff_ge {x} e1 SUBSET xfan (x,V,E)
==> aff_gt {x} {v, u} INTER aff_ge {x} e1={}`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC 
THEN REWRITE_TAC[SET_RULE`~(A={})<=> ?y. y IN A`]
THEN MATCH_MP_TAC exists_in_aff_gt
THEN ASM_REWRITE_TAC[]]);;

let properties_edges_eq_fan=prove(`!e:A-> bool v:A u:A.
FINITE e /\ ~(e={v,u}) /\ ~(v=u)  /\ CARD e=2  ==>  ~(v IN e)\/ ~(u IN e)`,
REPEAT GEN_TAC THEN STRIP_TAC
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC "YEU")
THEN ONCE_REWRITE_TAC[SET_RULE`~A \/ ~B<=> ~(A/\B)`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`v IN e /\ u IN e ==> {v:A,u} SUBSET e`)
THEN RESA_TAC
THEN MRESA_TAC CARD_2_FAN[`v:A`;`u:A`]
THEN REMOVE_THEN "YEU" MP_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN STRIP_TAC
THEN MRESA_TAC CARD_SUBSET_EQ[`{v,u:A}`;`e:A->bool`]);;

let condition_not_intersection_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 ds e1:real^3->bool e2:real^3->bool.
FAN(x,V,E) /\ v IN V /\ u IN V /\ ~collinear {x,v,u}
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ aff_gt {x} {v,u} SUBSET dartset_leads_into_fan x V E ds
/\ e1 IN E /\ e2 = {v, u}
==>aff_ge {x} e1 INTER aff_ge {x} e2 = aff_ge {x} (e1 INTER e2)`,
REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC aff_ge_eq_aff_gt_union_aff_ge[`(x:real^3)`;`(v:real^3)`;` (u:real^3)`]
THEN MRESA_TAC dartset_leads_into_subset_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` ds:real^3#real^3#real^3#real^3->bool`]
THEN MP_TAC(SET_RULE`dartset_leads_into_fan x V E ds SUBSET yfan (x,V,E)/\ aff_gt {x} {v, u} SUBSET dartset_leads_into_fan x V E ds==> aff_gt {x} {v, u} SUBSET yfan (x:real^3,V,E)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[yfan; SET_RULE`A SUBSET (:real^3) DIFF B <=> A INTER B={}`;]
THEN DISCH_TAC
THEN SUBGOAL_THEN`aff_ge {x} e1 SUBSET xfan(x:real^3,V:real^3->bool,E)` ASSUME_TAC
THENL[
REWRITE_TAC[xfan;SUBSET;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC `e1:real^3->bool`
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN SET_TAC[IN];
MP_TAC(SET_RULE`aff_gt {x:real^3} {v, u} INTER xfan (x,V:real^3->bool,E) = {}/\aff_ge {x} e1 SUBSET xfan (x,V,E)
==> aff_gt {x} {v, u} INTER aff_ge {x} e1={}`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`
aff_gt {x:real^3} {v, u} INTER aff_ge {x} e1={}==>
aff_ge {x} e1 INTER (aff_gt {x} {v, u} UNION aff_ge {x} {v} UNION aff_ge {x} {u})
= (aff_ge {x} e1 INTER aff_ge {x} {v}) UNION (aff_ge {x} e1 INTER aff_ge {x} {u})`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`e1 IN E==> e1 IN E UNION {{v:real^3} | v IN V}`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`v:real^3 IN V==> {v} IN E UNION {{v} | v IN V}`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`u:real^3 IN V==> {u} IN E UNION {{v} | v IN V}`)
THEN RESA_TAC
THEN FIND_ASSUM MP_TAC(`FAN(x:real^3,V,E)`)
THEN REWRITE_TAC[FAN;fan7]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th [`e1:real^3->bool`;`{v:real^3}`]THEN ASSUME_TAC th)
THEN POP_ASSUM(fun th-> MRESA_TAC th [`e1:real^3->bool`;`{u:real^3}`])
THEN MRESA_TAC condition_not_edge_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;` u:real^3`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN FIND_ASSUM MP_TAC (`graph (E:(real^3->bool)->bool)`)
THEN REWRITE_TAC[GRAPH]
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> MRESAL1_TAC th `e1:real^3->bool`[HAS_SIZE])
THEN MRESA_TAC th3[`v:real^3`;`u:real^3`;`x:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={C,A,B}`]
THEN RESA_TAC
THEN MRESA_TAC properties_edges_eq_fan[`e1:real^3->bool`; `v:real^3`;`u:real^3`]
THENL[MP_TAC(SET_RULE`~(v IN e1)==> e1 INTER {v:real^3}={} /\ e1 INTER {v, u}=e1 INTER {u}`)
THEN RESA_TAC
THEN ASSUME_TAC(SET_RULE`{} SUBSET e1 INTER {u}:real^3->bool`)
THEN MP_TAC(SET_RULE`DISJOINT {v, u} {x:real^3}==> DISJOINT {x} (e1 INTER {u})`)
THEN RESA_TAC
THEN MRESA_TAC AFF_GE_MONO_RIGHT[`{x:real^3}`;`{}:real^3->bool`;`e1 INTER {u}:real^3->bool`]
THEN POP_ASSUM MP_TAC THEN SET_TAC[];
MP_TAC(SET_RULE`~(u IN e1)==> e1 INTER {u:real^3}={} /\ e1 INTER {v, u}=e1 INTER {v}`)
THEN RESA_TAC
THEN ASSUME_TAC(SET_RULE`{} SUBSET e1 INTER {v}:real^3->bool`)
THEN MP_TAC(SET_RULE`DISJOINT {v, u} {x:real^3}==> DISJOINT {x} (e1 INTER {v})`)
THEN RESA_TAC
THEN MRESA_TAC AFF_GE_MONO_RIGHT[`{x:real^3}`;`{}:real^3->bool`;`e1 INTER {v}:real^3->bool`]
THEN POP_ASSUM MP_TAC THEN SET_TAC[]]]);;

let exists_edge_fully_surround_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) w:real^3.
FAN(x,V,E) /\ w IN V
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
==> ?v. {w,v} IN E /\ v IN V`,
REPEAT STRIP_TAC
THEN POP_ASSUM (fun th-> MRESA1_TAC th`w:real^3`)
THEN SUBGOAL_THEN`~(set_of_edge (w:real^3) V E={})`ASSUME_TAC
THENL[ POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC "YEU") THEN STRIP_TAC THEN REMOVE_THEN "YEU" MP_TAC
THEN ASM_REWRITE_TAC[CARD_CLAUSES] THEN ARITH_TAC;
POP_ASSUM MP_TAC 
THEN REWRITE_TAC[SET_RULE`~(A={})<=> ?v. v IN A`; set_of_edge;IN_ELIM_THM]]);;



let condition_not_intersection_point_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 ds:real^3#real^3#real^3#real^3->bool e1:real^3->bool w:real^3.
FAN(x,V,E) /\ v IN V /\ u IN V /\ ~collinear {x,v,u}
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ aff_gt {x} {v,u} SUBSET dartset_leads_into_fan x V E ds
/\ w IN V /\ e1 = {v, u}
==>aff_ge {x} e1 INTER aff_ge {x} {w} = aff_ge {x} (e1 INTER {w})`,
REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC aff_ge_eq_aff_gt_union_aff_ge[`(x:real^3)`;`(v:real^3)`;` (u:real^3)`]
THEN MRESA_TAC dartset_leads_into_subset_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` ds:real^3#real^3#real^3#real^3->bool`]
THEN MP_TAC(SET_RULE`dartset_leads_into_fan x V E ds SUBSET yfan (x,V,E)/\ aff_gt {x} {v, u} SUBSET dartset_leads_into_fan x V E ds==> aff_gt {x} {v, u} SUBSET yfan (x:real^3,V,E)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[yfan; SET_RULE`A SUBSET (:real^3) DIFF B <=> A INTER B={}`;]
THEN DISCH_TAC
THEN SUBGOAL_THEN`aff_ge {x} {w} SUBSET xfan(x:real^3,V:real^3->bool,E)` ASSUME_TAC
THENL[
REWRITE_TAC[xfan;SUBSET;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN MRESA_TAC exists_edge_fully_surround_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` w:real^3`]
THEN EXISTS_TAC `{w,v'}:real^3->bool`
THEN ASM_REWRITE_TAC[]
THEN REMOVE_ASSUM_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (v':real^3)`;
` (w:real^3)`]
THEN MRESA_TAC aff_ge_eq_aff_gt_union_aff_ge[`(x:real^3)`;`(w:real^3)`;` (v':real^3)`]
THEN ASM_TAC THEN SET_TAC[IN];
MP_TAC(SET_RULE`aff_gt {x:real^3} {v, u} INTER xfan (x,V:real^3->bool,E) = {}/\aff_ge {x} {w} SUBSET xfan (x,V,E)
==> aff_gt {x} {v, u} INTER aff_ge {x} {w}={}`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`
aff_gt {x:real^3} {v, u} INTER aff_ge {x} {w}={}==>
(aff_gt {x} {v, u} UNION aff_ge {x} {v} UNION aff_ge {x} {u}) INTER aff_ge {x} {w} 
= (aff_ge {x} {w} INTER aff_ge {x} {v}) UNION (aff_ge {x} {w} INTER aff_ge {x} {u})`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`w IN V==> {w} IN E UNION {{v:real^3} | v IN V}`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`v:real^3 IN V==> {v} IN E UNION {{v} | v IN V}`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`u:real^3 IN V==> {u} IN E UNION {{v} | v IN V}`)
THEN RESA_TAC
THEN FIND_ASSUM MP_TAC(`FAN(x:real^3,V,E)`)
THEN REWRITE_TAC[FAN;fan7]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th [`{w}:real^3->bool`;`{v:real^3}`]THEN ASSUME_TAC th)
THEN POP_ASSUM(fun th-> MRESA_TAC th [`{w}:real^3->bool`;`{u:real^3}`])
THEN MRESA_TAC th3[`v:real^3`;`u:real^3`;`x:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={C,A,B}`]
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(v=u:real^3)==> ({v, u} INTER {w}= {w} INTER {v} /\ {w} INTER {u}={}) \/ ({v, u} INTER {w}= {w} INTER {u} /\ {w} INTER {v}={})`)
THEN RESA_TAC
THENL[ ASSUME_TAC(SET_RULE`{} SUBSET {w} INTER {v}:real^3->bool`)
THEN MP_TAC(SET_RULE`DISJOINT {v, u} {x:real^3}==> DISJOINT {x} ({w} INTER {v})`)
THEN RESA_TAC
THEN MRESA_TAC AFF_GE_MONO_RIGHT[`{x:real^3}`;`{}:real^3->bool`;`{w} INTER {v}:real^3->bool`]
THEN POP_ASSUM MP_TAC THEN SET_TAC[];
ASSUME_TAC(SET_RULE`{} SUBSET {w} INTER {u}:real^3->bool`)
THEN MP_TAC(SET_RULE`DISJOINT {v, u} {x:real^3}==> DISJOINT {x} ({w} INTER {u})`)
THEN RESA_TAC
THEN MRESA_TAC AFF_GE_MONO_RIGHT[`{x:real^3}`;`{}:real^3->bool`;`{w} INTER {u}:real^3->bool`]
THEN POP_ASSUM MP_TAC THEN SET_TAC[]]]);;


let DWWUTKW=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 ds.
FAN(x,V,E) /\ v IN V /\ u IN V /\ ~collinear {x,v,u}
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ aff_gt {x} {v,u} SUBSET dartset_leads_into_fan x V E ds
/\ E1=E UNION {{v,u}}
==> FAN (x,V,E1)`,
REPEAT STRIP_TAC
THEN FIND_ASSUM MP_TAC`FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN;fan1;fan2;fan6;fan7]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"YEU")
THEN MRESA_TAC th3[`v:real^3`;`u:real^3`;`x:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={C,A,B}`]
THEN RESA_TAC
THEN MRESA_TAC add_edge_graph_fan[`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`]
THEN MRESA_TAC garph_add_edge_is_garph[`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`]
THEN MRESA_TAC add_edge_into_collinear_fan[`x:real^3`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`]
THEN REWRITE_TAC[UNION;IN_ELIM_THM;IN_SING]
THEN REPEAT GEN_TAC
THEN STRIP_TAC
THENL[ REMOVE_THEN"YEU" (fun th-> MRESA_TAC th[`e1:real^3->bool`;`e2:real^3->bool`])
THEN POP_ASSUM MATCH_MP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN SET_TAC[];

MRESA_TAC condition_not_intersection_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`; `ds:real^3#real^3#real^3#real^3->bool`;`e1:real^3->bool`;`e2:real^3->bool`];

REMOVE_THEN"YEU" (fun th-> MRESA_TAC th[`e1:real^3->bool`;`e2:real^3->bool`])
THEN POP_ASSUM MATCH_MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN SET_TAC[];

MRESA_TAC condition_not_intersection_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`; `ds:real^3#real^3#real^3#real^3->bool`;`e2:real^3->bool`;`e1:real^3->bool`]
THEN ASM_REWRITE_TAC[SET_RULE`A INTER B= B INTER A`];

ASM_REWRITE_TAC[SET_RULE`A INTER A=A`];

MRESA_TAC condition_not_intersection_point_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`; `ds:real^3#real^3#real^3#real^3->bool`;`e1:real^3->bool`;`v':real^3`];

REMOVE_THEN"YEU" (fun th-> MRESA_TAC th[`e2:real^3->bool`;`e1:real^3->bool`])
THEN ASM_REWRITE_TAC[SET_RULE`A INTER B=B INTER A`]
THEN POP_ASSUM MATCH_MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN SET_TAC[];

MRESA_TAC condition_not_intersection_point_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`; `ds:real^3#real^3#real^3#real^3->bool`;`e2:real^3->bool`;`v':real^3`]
THEN ASM_REWRITE_TAC[SET_RULE`A INTER B=B INTER A`];

REMOVE_THEN"YEU" (fun th-> MRESA_TAC th[`e1:real^3->bool`;`e2:real^3->bool`])
THEN POP_ASSUM MATCH_MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN SET_TAC[]]);;



(**************JUTSTKG*******************)


let exists_point_in_component_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool.
U IN topological_component_yfan (x,V,E)
==> ?z. z IN U`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[topological_component_yfan;IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`y:real^3`
THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_SET;IN_ELIM_THM]
THEN EXISTS_TAC`{y:real^3}`
THEN ASM_REWRITE_TAC[CONNECTED_SING;IN_SING]
THEN ASM_TAC THEN SET_TAC[]);;



let nonsetedge_fully_surround_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool).
(!v. v IN V==>CARD (set_of_edge v V E) >1)/\ FAN(x,V,E)
==> ~(E={})`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[]
THEN POP_ASSUM (fun th-> MP_TAC th THEN REWRITE_TAC[FAN;fan1]  THEN STRIP_TAC THEN ASSUME_TAC(th))
THEN FIND_ASSUM MP_TAC(`~((V:real^3->bool) SUBSET {})`)
THEN REWRITE_TAC[SET_RULE`~(V SUBSET {})<=> ?w. w IN V`]
THEN STRIP_TAC
THEN MRESA_TAC exists_edge_fully_surround_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`w:real^3`]
THEN ASM_TAC THEN SET_TAC[]);;




let exists_point_notx_in_xfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool).
FAN(x,V,E) /\ ~(E={})
==> ?v. v IN xfan(x,V,E) /\ ~(v=x)`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[]
THEN REWRITE_TAC[SET_RULE`~(A={})<=> ?y. y IN A`;xfan;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"YEU")
THEN MRESA_TAC expand_edge_graph_fan [`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(y:real^3->bool)`]
THEN REMOVE_THEN "YEU" MP_TAC
THEN RESA_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`v:real^3`]
THEN MRESA_TAC point_in_aff_ge[`(x:real^3)`;`(v:real^3)`;`(w:real^3)`]
THEN EXISTS_TAC `v:real^3`
THEN ASM_REWRITE_TAC[]
THEN EXISTS_TAC `y:real^3->bool`
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN SET_TAC[]);;


let x_in_xfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool).
FAN(x,V,E) /\ ~(E={})
==> x IN xfan(x,V,E)`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[]
THEN REWRITE_TAC[SET_RULE`~(A={})<=> ?y. y IN A`;xfan;IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC `y:real^3->bool`
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"YEU")
THEN MRESA_TAC expand_edge_graph_fan [`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(y:real^3->bool)`]
THEN REMOVE_THEN "YEU" MP_TAC
THEN RESA_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`v:real^3`]
THEN MRESA_TAC point_in_aff_ge[`(x:real^3)`;`(v:real^3)`;`(w:real^3)`]
THEN ASM_TAC THEN SET_TAC[]);;


let xfan_notempty_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool).
FAN(x,V,E) /\ ~(E={})
==> ~(xfan(x,V,E)={})`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[]
THEN REWRITE_TAC[SET_RULE`~(A={})<=> ?y. y IN A`]
THEN EXISTS_TAC `x:real^3`
THEN ASM_TAC
THEN SET_TAC[x_in_xfan]);;







let xfan_closed_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool).
FAN(x,V,E) 
==> closed (xfan(x,V,E))`,

REPEAT STRIP_TAC
THEN REWRITE_TAC[xfan;SET_RULE`{v | ?e. E e /\ v IN aff_ge {x} e}= UNIONS{y | ?x'. x' IN E /\ y = aff_ge {x} x'}`]
THEN MATCH_MP_TAC CLOSED_UNIONS 
THEN MRESA_TAC set_edges_is_finite_fan[`x:real^3`;`V:real^3->bool`;`E:(real^3->bool)->bool`]
THEN MRESAL_TAC FINITE_IMAGE[`(\e:real^3->bool. aff_ge {x:real^3} e)`;`E:(real^3->bool)->bool`][IMAGE;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"YEU") THEN DISCH_TAC
THEN MRESA_TAC expand_edge_graph_fan [`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(x':real^3->bool)`]
THEN REMOVE_THEN "YEU"MP_TAC THEN RESA_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`v:real^3`]
THEN MRESA_TAC closed_aff_ge_1_2[`x:real^3`;`v:real^3`;`w:real^3`]);;

let topological_component_subset_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool.
U IN topological_component_yfan (x,V,E)
==> U SUBSET yfan(x,V,E)`,
REWRITE_TAC[topological_component_yfan;IN_ELIM_THM;]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_SUBSET]);;








let aff_gt_connect_bound_not_inter_edges_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 y:real^3 z:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ DISJOINT {x} {y,z}
/\ (!t. &0< t /\ t< &1==>   (&1-t)%y+t%z IN yfan (x,V,E))
==>
aff_gt {x} {y,z} INTER aff_ge {x} {v,u}={}`,

REWRITE_TAC[SET_RULE`A={}<=> ~(?w. w IN A)`;INTER;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"YEU")
THEN REPEAT STRIP_TAC
THEN MRESA_TAC scale_in_edges_fan[`x:real^3`;`y:real^3`;`z:real^3`;`w:real^3`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MP_TAC(REAL_ARITH`&0<a==> &0 <= a`) THEN RESA_TAC
THEN MRESA_TAC scale_aff_ge_fan[`x:real^3`;`v:real^3`;`u:real^3`]
THEN POP_ASSUM (fun th-> MRESAL_TAC th[`w:real^3`;`a:real`][VECTOR_ARITH`(A+B-C)+C=A+B:real^3`])
THEN MRESA_TAC in_aff_gt_1_2[`x:real^3`;`y:real^3`;`z:real^3 `;`t:real`]
THEN REMOVE_THEN "YEU" (fun th-> MRESAL1_TAC th`t:real`[yfan])
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[SET_RULE`~((&1 - t) % y + t % z IN (:real^3) DIFF xfan (x,V,E))<=>(&1 - t) % y + t % z IN xfan (x,V,E)`;xfan;IN_ELIM_THM]
THEN EXISTS_TAC`{v,u}:real^3->bool`
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN SET_TAC[IN]);;







let aff_gt_connect_bound_subset_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) y:real^3 z:real^3.
FAN(x,V,E) /\ DISJOINT {x} {y,z} 
/\ (!t. &0< t /\ t< &1==>   (&1-t)%y+t%z IN yfan (x,V,E))
==> aff_gt {x} {y,z} SUBSET yfan (x,V,E)`,

REPEAT STRIP_TAC
THEN
REWRITE_TAC[yfan;xfan;SET_RULE`aff_gt {x} {y, z} SUBSET
         (:real^3) DIFF {v | ?e. E e /\ v IN aff_ge {x} e}
<=> (!e. e IN E ==> aff_gt {x} {y,z} INTER aff_ge {x} e={})`]
THEN REPEAT STRIP_TAC
THEN MRESA_TAC expand_edge_graph_fan [`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(e:real^3->bool)`]
THEN MRESA_TAC aff_gt_connect_bound_not_inter_edges_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;` w:real^3`;`y:real^3`;`z:real^3`]
THEN POP_ASSUM MATCH_MP_TAC
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM(th)] ));;

let sym_line1_fan=prove(`!x y z:real^N. x IN aff {y, z} /\ ~(x=y)
==> z IN aff {x,y}`,
REPEAT GEN_TAC
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN ASM_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(v= &0)\/ ~(v= &0)`)
THENL[ ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN RESA_TAC
THEN REDUCE_VECTOR_TAC
THEN RESA_TAC
THEN SET_TAC[];
MP_TAC(ISPEC`(v:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`inv(v:real)`
THEN EXISTS_TAC`-- inv(v:real) *u`
THEN ASM_REWRITE_TAC[REAL_ARITH` inv v + --inv v * u = inv v * (v+ &1- (u+v)) `;REAL_ARITH`A+ &1 - &1= A`;VECTOR_ARITH`inv v % (u % y + v % z) + (--inv v * u) % y=(inv v * v) % z `]
THEN REDUCE_VECTOR_TAC]);;


let POINT_IN_LINE=prove(`!x y:real^N. x IN  aff {x,y}`,
REPEAT GEN_TAC
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC`&0`
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC);;
let POINT_IN_LINE1=prove(`!x y:real^N. y IN  aff {x,y}`,
REPEAT GEN_TAC
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1`
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC);;

let AFFINE_HULL_AFFINE_EQ =prove(`!s:real^N->bool. affine hull (affine hull s)= affine  hull s`,
STRIP_TAC THEN MATCH_MP_TAC AFFINE_HULLS_EQ 
THEN ASSUME_TAC(ISPEC `s:real^N->bool` AFFINE_AFFINE_HULL)
THEN MRESA1_TAC AFFINE_HULL_EQ`affine hull s:real^N->bool`
THEN MRESA_TAC HULL_SUBSET[`affine:(real^N->bool)->bool`;` s:real^N->bool`;]
THEN SET_TAC[]);;
let sym_line0_fan=prove( `!x y z:real^N. x IN aff {y, z} /\ DISJOINT {x} {y,z}
==> aff {x,z} SUBSET aff {x,y}`,
REPEAT GEN_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`DISJOINT {x} {y,z}==> ~(x=y:real^N)`)
THEN RESA_TAC
THEN MRESA_TAC sym_line1_fan[`x:real^N`;`y:real^N`;`z:real^N`]
THEN MP_TAC(SET_RULE`x IN aff {x, y} /\ z IN aff {x, y} ==> {x, z:real^N} SUBSET aff {x, y}`)
THEN REWRITE_TAC[POINT_IN_LINE;]
THEN RESA_TAC
THEN MRESA_TAC HULL_MONO[`affine:(real^N->bool)->bool`;` {x, z:real^N}`;`aff {x, y:real^N}`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[aff;AFFINE_HULL_AFFINE_EQ ]);;

let sym_line_fan=prove(`!x y z:real^N. x IN aff {y, z} /\ DISJOINT {x} {y,z}
==> aff {x,z} = aff {x,y}`,
REPEAT STRIP_TAC
THEN MRESA_TAC sym_line0_fan[`x:real^N`;`y:real^N`;`z:real^N`]
THEN MRESA_TAC sym_line0_fan[`x:real^N`;`z:real^N`;`y:real^N`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN POP_ASSUM MP_TAC
THEN SET_TAC[]);;
let sym_line01_fan=prove(`!x y z:real^N. x IN aff {y, z} /\ DISJOINT {y} {x,z}
==> aff {y,x} SUBSET aff {y,z}`,
REPEAT GEN_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`y IN aff {y, z} /\ x IN aff {y, z} ==> {y,x:real^N} SUBSET aff {y,z}`)
THEN ASM_REWRITE_TAC[POINT_IN_LINE;]
THEN STRIP_TAC
THEN MRESA_TAC HULL_MONO[`affine:(real^N->bool)->bool`;` {y,x:real^N}`;`aff { y,z:real^N}`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[aff;AFFINE_HULL_AFFINE_EQ ]);;

let sym_line02_fan=prove(`!x y z:real^N. x IN aff {y, z} /\ DISJOINT {y} {x,z}
==> aff {y,z} SUBSET aff {y,x}`,
REPEAT GEN_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`DISJOINT {y} {x,z}==> ~(x=y:real^N)`)
THEN RESA_TAC
THEN MRESA_TAC sym_line1_fan[`x:real^N`;`y:real^N`;`z:real^N`]
THEN MP_TAC(SET_RULE`y IN aff {y, x} /\ z IN aff {y, x} ==> {y,z:real^N} SUBSET aff {y,x}`)
THEN ASM_REWRITE_TAC[POINT_IN_LINE;]
THEN STRIP_TAC
THEN MRESA_TAC HULL_MONO[`affine:(real^N->bool)->bool`;` {y,z:real^N}`;`aff { y,x:real^N}`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[aff;AFFINE_HULL_AFFINE_EQ ]
THEN POP_ASSUM MP_TAC
THEN ASSUME_TAC(SET_RULE`{y,x}={x,y}`)
THEN POP_ASSUM (fun th-> ASM_REWRITE_TAC[th;])
THEN REWRITE_TAC[aff]
THEN RESA_TAC);;


let sym_line_fan0=prove(`!x y z:real^N. x IN aff {y, z} /\ DISJOINT {x} {y,z} /\ DISJOINT {y} {x,z}
==> aff {x,z} = aff {y,z}`,
REPEAT STRIP_TAC
THEN MRESA_TAC sym_line_fan[`x:real^N`;`y:real^N`;`z:real^N`]
THEN SUBGOAL_THEN `y IN aff {x,z:real^N}` ASSUME_TAC
THENL[ASM_REWRITE_TAC[POINT_IN_LINE1];
MRESA_TAC sym_line_fan[`y:real^N`;`x:real^N`;`z:real^N`]
THEN ASSUME_TAC(SET_RULE`{x,y}={y,x:real^N}`)
THEN ASM_REWRITE_TAC[]]);;

let sym_line_fan1=prove(`!x y z:real^N. x IN aff {y, z} /\ DISJOINT {y} {x,z}
==> aff {y,z} = aff {y,x}`,
REPEAT STRIP_TAC
THEN MRESA_TAC sym_line01_fan[`x:real^N`;`y:real^N`;`z:real^N`]
THEN MRESA_TAC sym_line02_fan[`x:real^N`;`y:real^N`;`z:real^N`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[]);;

let aff_ge_1_1_subset_aff_fan=prove(`!x y z:real^3. ~(y=z) /\ x IN aff_ge {y} {z} ==> x IN aff {y,z} `,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC AFF_GE_1_1[`y:real^3`;`z:real^3`][IN_ELIM_THM]
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN ASM_REWRITE_TAC[]);;


let exists_point_notxin_convex_in_xfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) z:real^3.
FAN(x,V,E)  /\ ~(x=z) /\ ~(E={})
==> ?v. v IN xfan(x,V,E) /\ ~(x IN convex hull{v,z})`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[]
THEN REWRITE_TAC[SET_RULE`~(A={})<=> ?y. y IN A`;xfan;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"YEU")
THEN MRESA_TAC expand_edge_graph_fan [`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(y:real^3->bool)`]
THEN REMOVE_THEN "YEU" MP_TAC
THEN RESA_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`v:real^3`]
THEN MRESA_TAC point_in_aff_ge[`(x:real^3)`;`(v:real^3)`;`(w:real^3)`]
THEN DISJ_CASES_TAC(SET_RULE`~(x IN convex hull{v,z:real^3})\/ (x IN convex hull{v,z})`)
THENL[
EXISTS_TAC `v:real^3`
THEN ASM_REWRITE_TAC[]
THEN EXISTS_TAC `y:real^3->bool`
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN SET_TAC[];
EXISTS_TAC `w:real^3`
THEN STRIP_TAC
THENL[
EXISTS_TAC `y:real^3->bool`
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN SET_TAC[];
MRESA1_TAC CONVEX_HULL_SUBSET_AFFINE_HULL`{v,z:real^3}`
THEN MP_TAC(SET_RULE`x IN convex hull {v, z:real^3}/\ convex hull {v, z} SUBSET affine hull {v, z}
==> x IN affine hull {v, z}`)
THEN RESA_TAC
THEN STRIP_TAC
THEN MRESA1_TAC CONVEX_HULL_SUBSET_AFFINE_HULL`{w,z:real^3}`
THEN MP_TAC(SET_RULE`x IN convex hull {w, z:real^3}/\ convex hull {w, z} SUBSET affine hull {w, z}
==> x IN affine hull {w, z}`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(x=z) /\ ~(x=v) /\ ~(x=w)==> DISJOINT {x:real^3} {v,z} /\ DISJOINT{x} {w,z}`) THEN RESA_TAC
THEN MRESAL_TAC sym_line_fan[`x:real^3`;`v:real^3`;`z:real^3`][aff]
THEN MRESAL_TAC sym_line_fan[`x:real^3`;`w:real^3`;`z:real^3`][aff]
THEN MRESA_TAC POINT_IN_LINE1[`x:real^3` ;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[aff]
THEN POP_ASSUM (fun th -> REWRITE_TAC[SYM(th)])
THEN ASM_REWRITE_TAC[SYM aff]]]);;


let notempty_xfan_inter_segment_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) z:real^3 v:real^3.
FAN(x,V,E) /\ v IN xfan(x,V,E)
==> ~(xfan(x,V,E) INTER segment[v,z] ={})`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[SET_RULE`~(A={})<=> ?y. y IN A`;IN_ELIM_THM;INTER;segment]
THEN EXISTS_TAC`v:real^3`
THEN ASM_REWRITE_TAC[]
THEN EXISTS_TAC`&0`
THEN REWRITE_TAC[REAL_ARITH`&1- &0= &1 /\ &0<= &0 /\ &0<= &1`]
THEN REDUCE_VECTOR_TAC);;


let xfan_inter_segment_closed_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) z:real^3 v:real^3.
FAN(x,V,E) ==> closed(xfan(x,V,E) INTER segment[v,z])`,
REPEAT STRIP_TAC
THEN MATCH_MP_TAC CLOSED_INTER
THEN POP_ASSUM MP_TAC 
THEN REWRITE_TAC[CLOSED_SEGMENT;xfan_closed_fan]);;


let point_in_yfan_not_x_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool z:real^3.
FAN(x,V,E) /\ ~(E={})
/\ U IN topological_component_yfan (x,V,E)
/\ z IN U
==> ~(x=z)`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC topological_component_subset_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` U:real^3->bool`]
THEN MP_TAC(SET_RULE`U SUBSET yfan (x:real^3,V:real^3->bool,E)/\ z IN U ==> z IN yfan(x,V,E)`)
THEN RESA_TAC
THEN MRESA_TAC x_in_xfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[yfan]
THEN SET_TAC[]);;

let zpoint_in_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool z:real^3.
FAN(x,V,E) /\ ~(E={})
/\ U IN topological_component_yfan (x,V,E)
/\ z IN U
==> z IN yfan(x,V,E)`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC topological_component_subset_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` U:real^3->bool`]
THEN MP_TAC(SET_RULE`U SUBSET yfan (x:real^3,V:real^3->bool,E)/\ z IN U ==> z IN yfan(x,V,E)`)
THEN RESA_TAC
THEN SET_TAC[]);;



let segment_in_segment=prove(`!x y z:real^N. z IN segment [x,y]==>  (!t. &0<= t /\ t<= &1 ==> (&1-t) %z +t %y IN segment[x,y])`,
REWRITE_TAC[segment;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN EXISTS_TAC`(&1- t)* u+t:real`
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - t) % ((&1 - u) % x + u % y) + t % y =
 (&1 - ((&1 - t) * u + t)) % x + ((&1 - t) * u + t) % y:real^N`]
THEN STRIP_TAC
THENL[MATCH_MP_TAC REAL_LE_ADD
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= &1- t<=> t<= &1`];

REWRITE_TAC[REAL_ARITH`(&1 - t) * u + t <= &1<=> &0<= (&1 - t) * (&1-u)`]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= &1- t<=> t<= &1`]]);;

let connect_insidepoint_to_bound_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool z:real^3.
FAN(x,V,E) 
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ U IN topological_component_yfan (x,V,E)
/\ z IN U
==> ?y. ~(y=x)/\  y IN xfan(x,V,E) /\(!t. &0< t /\ t< &1==>   (&1-t)%y+t%z IN yfan(x,V,E))`,
REPEAT STRIP_TAC
THEN MRESA_TAC nonsetedge_fully_surround_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`]
THEN MRESA_TAC point_in_yfan_not_x_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;` z:real^3`]
THEN MRESA_TAC exists_point_notxin_convex_in_xfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` z:real^3`]
THEN EXISTS_TAC`closest_point (xfan (x:real^3,V:real^3->bool,E) INTER segment[v,z]) (z:real^3)`
THEN MRESA_TAC xfan_inter_segment_closed_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` z:real^3`;`v:real^3`]
THEN MRESA_TAC notempty_xfan_inter_segment_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` z:real^3`;`v:real^3`]
THEN MRESA_TAC CLOSEST_POINT_EXISTS[`xfan (x:real^3,V:real^3->bool,E) INTER segment[v,z]`;`(z:real^3)`]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"YEU")
THEN MP_TAC (SET_RULE`closest_point (xfan (x:real^3,V:real^3->bool,E) INTER segment [v,z:real^3]) z IN
      xfan (x,V,E) INTER segment [v,z]==> closest_point (xfan (x,V,E) INTER segment [v,z]) z IN
      xfan (x,V,E)/\ closest_point (xfan (x,V,E) INTER segment [v,z]) z IN segment [v,z]`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"EM") THEN DISCH_TAC
THEN POP_ASSUM (fun th-> MP_TAC th THEN ASSUME_TAC(th))
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SEGMENT_CONVEX_HULL]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`closest_point (xfan (x:real^3,V:real^3->bool,E) INTER convex hull {v, z}) z IN
 convex hull {v, z} /\ ~(x IN convex hull {v, z})==> ~(closest_point (xfan (x,V,E) INTER convex hull {v, z} ) z = x)`)
THEN RESA_TAC
THEN STRIP_TAC
THENL[ASM_REWRITE_TAC[SEGMENT_CONVEX_HULL];
ABBREV_TAC`y1=closest_point (xfan (x:real^3,V:real^3->bool,E) INTER segment [v,z]) z`
THEN SUBGOAL_THEN`!t. &0< t /\ t< &1==>dist (z,(&1 - t) % y1 + t % z)<dist (z,y1:real^3)` ASSUME_TAC
THENL[ REPEAT STRIP_TAC
THEN REWRITE_TAC[dist;VECTOR_ARITH`z - ((&1 - t) % y1 + t % z)=(&1 - t)%(z-y1)`]
THEN MP_TAC(ISPEC`&1- t:real`REAL_ABS_REFL)
THEN REWRITE_TAC[REAL_ARITH`&0<= &1-t <=> t<= &1`;NORM_MUL]
THEN MP_TAC(REAL_ARITH`t< &1==> t<= &1`) THEN RESA_TAC
THEN RESA_TAC
THEN MRESAL_TAC REAL_LT_RMUL[`&1-t:real`;`&1:real`;`norm (z - y1:real^3)`][REAL_ARITH`&1*A=A`;REAL_ARITH`&1-t< &1<=> &0< t`]
THEN POP_ASSUM MATCH_MP_TAC
THEN MATCH_MP_TAC(REAL_ARITH`&0<= A /\ ~(A= &0) ==> &0 < A`)
THEN REWRITE_TAC[NORM_POS_LE;NORM_EQ_0;VECTOR_ARITH`A-B= vec 0<=>A=B`]
THEN STRIP_TAC
THEN REMOVE_THEN "EM" MP_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN MRESA_TAC topological_component_subset_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`]
THEN MP_TAC(SET_RULE`z:real^3 IN U /\ U SUBSET yfan(x,V,E) ==> z IN yfan(x:real^3,V:real^3->bool,E)`)
THEN ASM_REWRITE_TAC[yfan]
THEN SET_TAC[];
POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"NHIEU")
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[yfan;DIFF;IN_ELIM_THM;SET_RULE`(&1 - t) % y1 + t % z IN (:real^3)`]
THEN STRIP_TAC
THEN MP_TAC (REAL_ARITH`&0<t /\ t< &1==> &0<= t /\ t:real<= &1`) THEN RESA_TAC
THEN MRESA_TAC segment_in_segment[`v:real^3`;`z:real^3`;`y1:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th`t:real`)
THEN MP_TAC(SET_RULE`(&1 - t) % y1 + t % z IN xfan (x:real^3,V:real^3->bool,E) /\ (&1 - t) % y1 + t % z IN segment [v,z]==> (&1 - t) % y1 + t % z IN xfan (x,V,E) INTER segment [v,z]`)
THEN RESA_TAC
THEN REMOVE_THEN "YEU"(fun th-> MRESA1_TAC th`(&1 - t) % y1 + t % z:real^3`)
THEN REMOVE_THEN "NHIEU"(fun th-> MRESA1_TAC th`t:real`)
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC]]);;

let in_topological_component_yfan_is_connected=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool.
U IN topological_component_yfan (x,V,E)
==> connected U`,
REWRITE_TAC[topological_component_yfan;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT]);;


let expand_element_in_topological_component_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool z:real^3.
FAN(x,V,E) 
/\ U IN topological_component_yfan (x,V,E)
/\ z IN U
==> U=connected_component (yfan(x,V,E)) z`,
REWRITE_TAC[topological_component_yfan;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC 
THEN MRESA_TAC CONNECTED_COMPONENT_EQ[`yfan(x:real^3, (V:real^3->bool) ,E)`;`y:real^3`;`z:real^3`]);;


let segmentsubset_aff_gt=prove(`!x y z w:real^N.
DISJOINT {x} {y,z}/\ w IN aff_gt {x} {y,z}
==> !t. &0<= t /\ t< &1 ==> (&1-t) %w+t%z IN aff_gt {x} {y,z}`,
REPEAT GEN_TAC THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC AFF_GT_1_2[`x:real^N`;`y:real^N`;`z:real^N`][IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN EXISTS_TAC`(&1-t)*t1:real`
THEN EXISTS_TAC`(&1-t)*t2:real`
THEN EXISTS_TAC`(&1-t)*t3+t:real`
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(&1 - t) % (t1 % x + t2 % y + t3 % z) + t % z =
 ((&1 - t) * t1) % x + ((&1 - t) * t2) % y + ((&1 - t) * t3 + t) % z:real^N`;REAL_ARITH`(&1 - t) * t1 + (&1 - t) * t2 + (&1 - t) * t3 + t=(&1 - t) * (t1 + t2 +t3) + t/\ (&1 - t) * &1 + t = &1`]
THEN STRIP_TAC
THENL[MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0< &1- t<=> t< &1`];

MATCH_MP_TAC (REAL_ARITH`&0<A /\ &0<=B==> &0< A+B`)
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0< &1- t<=> t< &1`]]);;
let point_in_aff_gt_in_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) y:real^3 z:real^3 w:real^3.
FAN(x,V,E) /\ DISJOINT {x} {y,z} /\ w IN aff_gt {x} {y,z}
/\ (!t. &0 < t /\ t < &1 ==> (&1 - t) % y + t % z IN yfan (x,V,E))
==> w IN yfan(x,V,E)`,
REPEAT STRIP_TAC
THEN MRESA_TAC aff_gt_connect_bound_subset_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`y:real^3`;`z:real^3`]
THEN ASM_TAC THEN SET_TAC[]);;



let segment_subset_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) y:real^3 z:real^3 w:real^3.
FAN(x,V,E) /\ DISJOINT {x} {y,z} /\ z IN yfan(x,V,E) /\ w IN aff_gt {x} {y,z}
/\ (!t. &0 < t /\ t < &1 ==> (&1 - t) % y + t % z IN yfan (x,V,E))
==> segment[w,z] SUBSET yfan(x,V,E)`,

REWRITE_TAC[segment;SUBSET;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN MRESA_TAC aff_gt_connect_bound_subset_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`y:real^3`;`z:real^3`]
THEN MRESA_TAC segmentsubset_aff_gt[`x:real^3`;`y:real^3`;`z:real^3`;`w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th`u:real`)
THEN POP_ASSUM MP_TAC
THEN MP_TAC(REAL_ARITH`u<= &1==> u< &1\/ u= &1`)
THEN RESA_TAC
THENL[
ASM_TAC THEN SET_TAC[];
ASM_REWRITE_TAC[VECTOR_ARITH`(&1 - &1) % w + &1 % z=z:real^3`]]);;

let exists_in_aff_gt_disjoint=prove(`!x:real^3  v:real^3 u:real^3.
DISJOINT {x} {v,u} ==> ?y:real^3. y IN aff_gt {x} {v, u}`,
REPEAT STRIP_TAC
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&0 % x+ &1 / &2 % v+ &1/ &2 %u:real^3 `
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1/ &2`
THEN EXISTS_TAC`&1/ &2`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC);;


let aff_gt_subset_component_y_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool y:real^3 z:real^3.
FAN(x,V,E) 
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ U IN topological_component_yfan (x,V,E)
/\ z IN U
/\ DISJOINT {x} {y,z}
/\ (!t. &0 < t /\ t < &1 ==> (&1 - t) % y + t % z IN yfan (x,V,E))
==> aff_gt {x} {y,z} SUBSET U`,
REPEAT STRIP_TAC
THEN MRESA_TAC expand_element_in_topological_component_yfan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`U:real^3->bool`;`z:real^3`]
THEN MRESA_TAC exists_in_aff_gt_disjoint[`x:real^3`;`y:real^3`;`z:real^3`]
THEN MRESA_TAC CONVEX_AFF_GT[`{x:real^3}`;`{y,z:real^3}`]
THEN MRESA1_TAC CONVEX_CONNECTED`aff_gt {x:real^3} {y,z:real^3}`
THEN MRESA_TAC aff_gt_connect_bound_subset_yfan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`y:real^3`;`z:real^3`]
THEN MRESA_TAC CONNECTED_COMPONENT_MAXIMAL[`yfan(x:real^3, (V:real^3->bool) ,E)`;`aff_gt {x:real^3} {y,z}`;`y':real^3`;]
THEN MATCH_MP_TAC(SET_RULE`aff_gt {x} {y, z} SUBSET connected_component (yfan (x,V,E)) y'/\ connected_component (yfan (x:real^3, (V:real^3->bool),E)) y'= connected_component (yfan (x,V,E)) z==> aff_gt {x} {y, z} SUBSET connected_component (yfan (x,V,E)) z`)
THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_EQ_EQ;]
THEN MRESA_TAC point_in_aff_gt_in_yfan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`y:real^3`;`z:real^3`;`y':real^3`]
THEN MRESA_TAC nonsetedge_fully_surround_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`]
THEN MRESA_TAC zpoint_in_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;`z:real^3`]
THEN MRESA_TAC CONNECTED_COMPONENT_OF_SUBSET [`segment [y',z:real^3]`;`yfan(x:real^3, (V:real^3->bool) ,E)`;`y':real^3`;`z:real^3`]
THEN POP_ASSUM MATCH_MP_TAC
THEN MRESA_TAC segment_subset_yfan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`y:real^3`;`z:real^3`;`y':real^3`]
THEN REWRITE_TAC[SEGMENT_CONVEX_HULL]
THEN MRESA1_TAC CONVEX_CONVEX_HULL`{y',z:real^3}`
THEN MRESA1_TAC CONVEX_CONNECTED `convex hull {y', z:real^3}`
THEN MRESA1_TAC CONNECTED_IFF_CONNECTED_COMPONENT`convex hull {y', z:real^3}`
THEN POP_ASSUM(fun th-> MRESA_TAC th[`y':real^3`;`z:real^3`])
THEN POP_ASSUM MATCH_MP_TAC
THEN MRESAL_TAC expansion1_convex_fan[`y':real^3`;`z:real^3`;`&0`][REAL_ARITH`&0<= &0 /\ &0<= &1`;VECTOR_ARITH`(&1- &0)%v+ &0 % u=v`]
THEN MRESAL_TAC expansion1_convex_fan[`y':real^3`;`z:real^3`;`&1`][REAL_ARITH`&0<= &1 /\ &1 <= &1`;VECTOR_ARITH`(&1- &1)%v+ &1 % u=u`]);;





let exists_connect_point_in_xfanto_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool z:real^3.
FAN(x,V,E) 
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ U IN topological_component_yfan (x,V,E)
/\ z IN U
==> ?y. ~(y=x) /\ y IN xfan(x,V,E) /\(!t. &0< t /\ t< &1==>   (&1-t)%y+t%z IN U)`,

REPEAT STRIP_TAC
THEN MRESA_TAC connect_insidepoint_to_bound_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`; `U:real^3->bool`;`z:real^3`]
THEN EXISTS_TAC `y:real^3`
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC nonsetedge_fully_surround_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`]
THEN MRESA_TAC point_in_yfan_not_x_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;` z:real^3`]
THEN MP_TAC(SET_RULE`~(x=z) /\ ~(y=x)==> DISJOINT {x} {y,z:real^3}`)
THEN RESA_TAC
THEN MRESA_TAC aff_gt_subset_component_y_fan [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;`z:real^3`]
THEN REPEAT STRIP_TAC
THEN MATCH_MP_TAC(SET_RULE`aff_gt {x:real^3} {y, z} SUBSET U /\ (&1 - t) % y + t % z IN aff_gt {x} {y, z}==> (&1 - t) % y + t % z IN U`)
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC AFF_GT_1_2[`(x:real^3)` ;` (y:real^3)`;`(z:real^3) `;][IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC `&1- t:real`
THEN EXISTS_TAC `t:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`&0< &1- t<=> t< &1`;REAL_ARITH`&0 + &1 - t + t = &1`]
THEN VECTOR_ARITH_TAC);;



let place_there_point_line_fan=prove(`!x:real^3 y:real^3 z:real^3.
~(x=y)/\ z IN aff {x,y}==> ?t:real. &0<t /\ t< &1 /\ (&1-t)%y+t % z IN aff_ge {x} {y}`,
REPEAT GEN_TAC
THEN REWRITE_TAC[aff; AFFINE_HULL_2;IN_ELIM_THM]
THEN RESA_TAC
THEN MRESAL_TAC AFF_GE_1_1[`x:real^3`;`y:real^3`][IN_ELIM_THM]
THEN DISJ_CASES_TAC(REAL_ARITH`&0<= v \/ v< &0`)
THENL[EXISTS_TAC`&1/ &2`
THEN ASM_REWRITE_TAC[REAL_ARITH`&0< &1/ &2 /\ &1/ &2< &1`]
THEN EXISTS_TAC`&1/ &2 *u`
THEN EXISTS_TAC`&1/ &2*(&1+v)`
THEN ASM_REWRITE_TAC[REAL_ARITH`&1/ &2 *u + &1/ &2 * (&1 + v)= &1/ &2 *(&1 +(u+v))`;REAL_ARITH`&1 / &2 * (&1 + &1) = &1`;VECTOR_ARITH`(&1 - &1 / &2) % y + &1 / &2 % (u % x + v % y) =
 (&1 / &2 * u) % x + (&1 / &2 * (&1 + v)) % y`]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= &1/ &2`]
THEN MATCH_MP_TAC REAL_LE_ADD
THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC;
EXISTS_TAC`inv u`
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`v< &0 <=> (u+v)< u `]
THEN STRIP_TAC
THEN MP_TAC(REAL_ARITH`&1< u==> &0< u /\ ~(u= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV `u:real`
THEN MRESA1_TAC REAL_INV_LT_1 `u:real`
THEN MRESA1_TAC REAL_MUL_LINV `u:real`
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC `&0`
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= &0 /\ &1 + &0 = &1/\ u+ &1 - &1 =u`;VECTOR_ARITH`(&1 - inv u) % y + inv u % (u % x + v % y)=(&1 - inv u*(u+ &1- (u+v))) % y + (inv u * u) % x /\ (&1 - &1) % y + &1 % x = &1 % x + &0 % y`]]);;



let aff_ge_1_1_subset_xfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) y:real^3.
FAN(x,V,E) 
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ y IN xfan(x,V,E) /\ ~(x=y)
==>  aff_ge {x} {y} SUBSET xfan(x,V,E)`,

REWRITE_TAC[xfan;IN_ELIM_THM;SUBSET]
THEN REPEAT STRIP_TAC
THEN MRESA_TAC expand_edge_graph_fan [`(x:real^3)`;` (V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(e:real^3->bool)`]
THEN POP_ASSUM MP_TAC
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[IN]
THEN RESA_TAC
THEN EXISTS_TAC`e:real^3->bool`
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC aff_ge_1_1_subset_aff_ge_fan[`x:real^3`;`v:real^3`;`w:real^3`;`y:real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)] THEN ASM_REWRITE_TAC[] THEN ASSUME_TAC(th))
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (v:real^3)`]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)] THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[IN]THEN ASM_REWRITE_TAC[] THEN ASSUME_TAC(th))
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN SET_TAC[]);;

let point_in_yfan_and_point_in_xfan_indepent_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool y:real^3 z:real^3.
FAN(x,V,E) 
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ U IN topological_component_yfan (x,V,E)
/\ z IN U
/\ y IN xfan(x,V,E) 
/\ ~(y=x) /\(!t. &0< t /\ t< &1==>   (&1-t)%y+t%z IN yfan (x,V,E))
==> ~collinear {x,y,z}`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"EM")
THEN MRESA_TAC nonsetedge_fully_surround_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`]
THEN MRESA_TAC zpoint_in_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;`z:real^3`]
THEN ASM_REWRITE_TAC[collinear_fan;SET_RULE`~(y=x)<=> ~(x=y)`]
THEN STRIP_TAC
THEN MRESA_TAC place_there_point_line_fan[`x:real^3`;`y:real^3`;`z:real^3`]
THEN MRESA_TAC aff_ge_1_1_subset_xfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`y:real^3`]
THEN MP_TAC(SET_RULE`(&1 - t) % y + t % z IN aff_ge {x:real^3} {y}/\ aff_ge {x} {y} SUBSET xfan (x,V:real^3->bool,E) ==> (&1 - t) % y + t % z IN  xfan (x,V,E)`)
THEN RESA_TAC
THEN REMOVE_THEN "EM"(fun th-> MRESA1_TAC th`t:real`)
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[yfan]
THEN POP_ASSUM MP_TAC
THEN SET_TAC[]);;
let permutes_4points_collinear=prove(`!x y z w:real^N.
~(x=y)/\ ~(x=z) /\ y IN aff {x,z}/\ ~collinear{x,y,w}==> ~collinear{x,z,w}`,
REWRITE_TAC[collinear_fan]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN MP_TAC(SET_RULE`~(x=y)/\ ~(x=z) ==> DISJOINT {x} {y, z:real^N}`)
THEN RESA_TAC
THEN MRESA_TAC sym_line_fan1[`y:real^N`;`x:real^N`;`z:real^N`]);;

let permutes_4points_collinear1=prove(`!x y z w:real^N.
~(x=y)/\ ~(x=z) /\  y IN aff {x,z}/\ ~collinear{x,z,w}==> ~collinear{x,y,w}`,
REWRITE_TAC[collinear_fan]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN MP_TAC(SET_RULE`~(x=y)/\ ~(x=z) ==>DISJOINT {x} {y, z:real^N}`)
THEN RESA_TAC
THEN MRESA_TAC sym_line01_fan[`y:real^N`;`x:real^N`;`z:real^N`]
THEN ASM_TAC  THEN SET_TAC[]);;

let in_aff_gt_eq_azim=prove(`!x y z w0 w1:real^3.
~(x=z) /\ y IN aff_gt {x} {z}==> azim x y w0 w1=azim x z w0 w1`,
REPEAT STRIP_TAC 
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC AFF_GT_1_1[`x:real^3`;`z:real^3`][IN_ELIM_THM;SET_RULE`DISJOINT {x} {z}<=> ~(x=z)`]
THEN ASM_TAC
THEN GEOM_ORIGIN_TAC `x:real^3`
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`u % (x + vec 0) + v%(x+ z)= (u+v)%x+v %z`;VECTOR_ARITH`(x + y = &1 % x + v % z)<=> y = v % z`]
THEN RESA_TAC
THEN ASM_TAC THEN SET_TAC[AZIM_SPECIAL_SCALE]);;
let no_origin_aff_ge_is_aff_gt=prove(`!x y z:real^3. 
~(x=y) /\ ~(x=z) /\ z IN aff_ge {x} {y}==> z IN aff_gt {x} {y}`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC AFF_GT_1_1[`x:real^3`;`y:real^3`][IN_ELIM_THM;SET_RULE`DISJOINT {x} {z}<=> ~(x=z)`]
THEN MRESAL_TAC AFF_GE_1_1[`x:real^3`;`y:real^3`][IN_ELIM_THM;SET_RULE`DISJOINT {x} {z}<=> ~(x=z)`]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
THEN MP_TAC(REAL_ARITH`&0<= t2==> t2= &0 \/ &0< t2`)
THEN RESA_TAC
THEN REDUCE_ARITH_TAC
THEN RESA_TAC
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN ASM_REWRITE_TAC[]);;


(*CASE 1*)



let exists_edge_component_yfan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (y:real^3).
 FAN(x,V,E) /\ (!v. v IN V==>CARD (set_of_edge v V E) >1) /\ v IN V
/\  ~(y IN set_of_edge v V E) 
==>
(?(w:real^3). (w IN (set_of_edge v V E)) /\
(!(w1:real^3). (w1 IN (set_of_edge v V E)) ==> azim1 x v y w <=  azim1 x v y w1))`,

(let lemma = prove
   (`!X:real->bool. 
          FINITE X /\ ~(X = {}) 
          ==> ?a. a IN X /\ !x. x IN X ==> a <= x`,
    MESON_TAC[INF_FINITE]) in

REPEAT STRIP_TAC
THEN MRESA_TAC exists_edge_fully_surround_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` v:real^3`]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` v':real^3`;`v:real^3`]
THEN MRESA_TAC FINITE_IMAGE [`azim1 (x:real^3) v y`;`(set_of_edge v (V:real^3->bool) (E:(real^3->bool)->bool))`]
THEN MP_TAC(SET_RULE`v' IN set_of_edge v V E==> ~(set_of_edge v (V:real^3->bool) (E:(real^3->bool)->bool)= {}) `)
THEN RESA_TAC
THEN MRESA_TAC IMAGE_EQ_EMPTY[`azim1 (x:real^3) v y`;`(set_of_edge v (V:real^3->bool) (E:(real^3->bool)->bool))`]
THEN MRESA1_TAC lemma`IMAGE (azim1 (x:real^3) v y) (set_of_edge v (V:real^3->bool) (E:(real^3->bool)->bool))
`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[IMAGE;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`x':real^3`
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"YEU")
THEN REPEAT STRIP_TAC
THEN SUBGOAL_THEN`(?x'. x' IN set_of_edge v V E /\ azim1 x v y w1 = azim1 x v y (x':real^3))`ASSUME_TAC
THENL[
EXISTS_TAC`w1:real^3`
THEN ASM_REWRITE_TAC[];
REMOVE_THEN "YEU" (fun th-> MRESA1_TAC th`azim1 x v y (w1:real^3)`)]));;



let v_subset_xfan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool).
FAN(x,V,E) /\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
==> V SUBSET xfan(x,V,E)`,
REWRITE_TAC[xfan;SUBSET;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN MRESA_TAC exists_edge_fully_surround_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` x':real^3`]
THEN EXISTS_TAC`{x',v:real^3}`
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` v:real^3`;`x':real^3`]
THEN MRESA_TAC point_in_aff_ge[`(x:real^3)`;`(x':real^3)`;`(v:real^3)`]
THEN ASM_TAC THEN SET_TAC[IN]);;



let set_of_edge_subset_edges=prove(`!(V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3).
 set_of_edge v V E SUBSET V`,
REWRITE_TAC[set_of_edge] THEN SET_TAC[]);;






let aff_ge_2_1_is_exists_point_inaff_ge_1_2=prove(`!x:real^3 y:real^3 z:real^3 w:real^3.
DISJOINT {x} {y,w} /\ DISJOINT {x,y} {w}/\ z IN aff_ge {x,y} {w}==> ?t. &0<t /\ t< &1 /\ (&1-t) %y+ t%z IN aff_ge {x} {y,w}`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC AFF_GE_2_1[`x:real^3`;`y:real^3`;`w:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN MRESAL_TAC AFF_GE_1_2[`x:real^3`;`y:real^3`;`w:real^3`][IN_ELIM_THM]
THEN DISJ_CASES_TAC(REAL_ARITH`&0<= t2 \/ t2 < &0`)
THENL[EXISTS_TAC`&1/ &2`
THEN ASM_REWRITE_TAC[REAL_ARITH`&0< &1/ &2/\ &1/ &2 < &1`]
THEN EXISTS_TAC`&1/ &2 * t1:real`
THEN EXISTS_TAC`&1/ &2 * (t2+ &1):real`
THEN EXISTS_TAC`&1/ &2 * t3:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`&1 / &2 * t1 + &1 / &2 * (t2 + &1) + &1 / &2 * t3 = &1/ &2 *(&1 +(t1+t2+t3))`;VECTOR_ARITH`(&1 - &1 / &2) % y + &1 / &2 % (t1 % x + t2 % y + t3 % w) =
 (&1 / &2 * t1) % x + (&1 / &2 * (t2 + &1)) % y + (&1 / &2 * t3) % w`;REAL_ARITH`&1/ &2 *(&1 + &1)= &1`]
THEN STRIP_TAC
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= &1/ &2`]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;

EXISTS_TAC`inv(&1 - t2):real`
THEN MP_TAC(REAL_ARITH`t2< &0 ==> &0< &1 -t2 /\ &1< &1 -t2 /\ &0<= &1 -t2 /\ ~(&1- t2= &0) `)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`&1-t2`
THEN MRESA1_TAC REAL_LE_INV`&1-t2`
THEN MRESA1_TAC REAL_MUL_LINV `&1- t2:real`
THEN MRESA1_TAC REAL_INV_LT_1`&1- t2`
THEN EXISTS_TAC`inv(&1 - t2)* t1:real`
THEN EXISTS_TAC`inv(&1 - t2)* t2 + &1 - inv(&1 - t2):real`
THEN EXISTS_TAC`inv(&1 - t2)* t3`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (&1 - t2) * t1 +
 (inv (&1 - t2) * t2 + &1 - inv (&1 - t2)) +
 inv (&1 - t2) * t3= inv (&1 - t2) * (t1 +t2+t3)+ &1 - inv (&1 - t2) /\ inv (&1 - t2) * &1 + &1 - inv (&1 - t2) = &1`;VECTOR_ARITH`(&1 - inv (&1 - t2)) % y + inv (&1 - t2) % (t1 % x + t2 % y + t3 % w) =
 (inv (&1 - t2) * t1) % x +
 (inv (&1 - t2) * t2 + &1 - inv (&1 - t2)) % y +
 (inv (&1 - t2) * t3) % w`;REAL_ARITH`inv (&1 - t2) * t2 + &1 - inv (&1 - t2)= &1 - inv (&1 - t2) *(&1-t2)`;REAL_ARITH`&0<= &1 - &1`]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[]]);;

let not_azim_points_in_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool y:real^3 z:real^3 u:real^3.
FAN(x,V,E) 
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ U IN topological_component_yfan (x,V,E)
/\ z IN U
/\ u IN V
/\ y IN aff_ge {x} {u}
/\ y IN xfan (x,V,E)
/\ ~(y=x)
 /\(!t. &0< t /\ t< &1==>   (&1-t)%y+t%z IN yfan (x,V,E))
==>(!(w1:real^3). (w1 IN (set_of_edge u V E)) ==> ~(azim x u z w1= &0))`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"YEU")
THEN DISCH_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w1:real^3)`;
` (u:real^3)`]
THEN POP_ASSUM MP_TAC
THEN RESA_TAC
THEN MRESA_TAC nonsetedge_fully_surround_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`]
THEN MRESA_TAC point_in_yfan_not_x_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;` z:real^3`]
THEN MRESA_TAC no_origin_aff_ge_is_aff_gt[`x:real^3`;`u:real^3`;`y:real^3`]
THEN MRESA_TAC in_aff_gt_eq_azim[`x:real^3`;`y:real^3`;`u:real^3`;`z:real^3`;`w1:real^3`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th)] THEN ASSUME_TAC(SYM(th)))
THEN MRESA_TAC point_in_yfan_and_point_in_xfan_indepent_fan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;` z:real^3`]
THEN MRESA_TAC aff_ge_1_1_subset_aff_fan[`y:real^3`;`x:real^3`;`u:real^3`]
THEN MRESA_TAC permutes_4points_collinear1[`x:real^3`;`y:real^3`;`u:real^3`;`w1:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`y:real^3`;`w1:real^3`]
THEN STRIP_TAC
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`y:real^3`;`z:real^3`;`w1:real^3`]
THEN MRESA_TAC aff_ge_2_1_is_exists_point_inaff_ge_1_2[`x:real^3`;`y:real^3`;`z:real^3`;`w1:real^3`]
THEN MRESA_TAC aff_ge_eq_aff_gt_union_aff_ge[`x:real^3`;`u:real^3`;`w1:real^3`]
THEN MP_TAC(SET_RULE`y IN aff_ge {x} {u}/\ aff_ge {x} {u, w1} =
      aff_gt {x} {u, w1} UNION aff_ge {x} {u} UNION aff_ge {x} {w1}==> y IN  aff_ge {x} {u, w1:real^3}`)
THEN POP_ASSUM(fun th -> GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MRESA_TAC aff_ge1_subset_aff_ge[`x:real^3`;`u:real^3`;`w1:real^3`;`y:real^3`]
THEN MP_TAC(SET_RULE`(&1 - t) % y + t % z IN aff_ge {x} {y, w1}
 /\ aff_ge {x} {y, w1:real^3} SUBSET aff_ge {x} {u, w1}==> (&1 - t) % y + t % z IN aff_ge {x} {u, w1}`)
THEN RESA_TAC
THEN SUBGOAL_THEN`aff_ge {x} {u,w1} SUBSET xfan(x:real^3,V:real^3->bool,E)` ASSUME_TAC
THENL[
ASM_REWRITE_TAC[xfan; SUBSET;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`{u,w1:real^3}`
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN SET_TAC[IN];
MP_TAC(SET_RULE`(&1 - t) % y + t % z IN aff_ge {x:real^3} {u, w1}/\ aff_ge {x} {u, w1} SUBSET xfan (x,V:real^3->bool,E)
==> (&1 - t) % y + t % z IN  xfan (x,V,E)`)
THEN RESA_TAC
THEN REMOVE_THEN "YEU"(fun th -> MRESA1_TAC th`t:real` )
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[yfan]
THEN SET_TAC[]]);;

let exists_edge_bounded_topological_component_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool y:real^3 z:real^3 u:real^3.
FAN(x,V,E) 
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ U IN topological_component_yfan (x,V,E)
/\ z IN U
/\ u IN V
/\ y IN aff_ge {x} {u}
/\ y IN xfan(x,V,E)
/\ ~(y=x)
/\(!t. &0< t /\ t< &1==>   (&1-t)%y+t%z IN yfan (x,V,E))
==> ?w. {u,w} IN E /\  z IN w_dart_fan x V E (x,u,w,sigma_fan x V E u w)`,

REPEAT STRIP_TAC
THEN MRESA_TAC nonsetedge_fully_surround_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`]
THEN MRESA_TAC zpoint_in_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;`z:real^3`]
THEN MRESA_TAC v_subset_xfan[`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`]
THEN MRESA_TAC set_of_edge_subset_edges[`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`u:real^3`]
THEN MP_TAC (SET_RULE`V SUBSET xfan (x,(V:real^3->bool),(E:(real^3->bool)->bool)) /\ set_of_edge u V E SUBSET V==> set_of_edge u V E SUBSET xfan (x,V,E)`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE` z IN yfan (x,(V:real^3->bool),(E:(real^3->bool)->bool)) /\ set_of_edge u V E SUBSET xfan (x,V,E)/\ yfan (x,V,E)= (:real^3) DIFF xfan (x,V,E)
==> ~(z IN set_of_edge u V E)`)
THEN ASM_REWRITE_TAC[yfan]
THEN STRIP_TAC
THEN MRESA_TAC exists_edge_component_yfan[`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (u:real^3)`; `(z:real^3)`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"YEU")
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`u:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN EXISTS_TAC`w:real^3`
THEN ASM_REWRITE_TAC[w_dart_fan]
THEN FIND_ASSUM MP_TAC(`(!v:real^3. v IN V==>CARD (set_of_edge v V E) >1)`)
THEN DISCH_TAC
THEN  POP_ASSUM(fun th->  MP_TAC(ISPEC`u:real^3` th))
THEN FIND_ASSUM MP_TAC(`(u:real^3) IN V`)
THEN DISCH_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[wedge;IN_ELIM_THM]
THEN MRESA_TAC point_in_yfan_and_point_in_xfan_indepent_fan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;` z:real^3`]
THEN MRESA_TAC aff_ge_1_1_subset_aff_fan[`y:real^3`;`x:real^3`;`u:real^3`]
THEN MRESA_TAC permutes_4points_collinear[`x:real^3`;`y:real^3`;`u:real^3`;`z:real^3`]
THEN MRESA_TAC not_azim_points_in_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` U:real^3->bool`;`y:real^3`;`z:real^3`;`u:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th`w:real^3`)
THEN MRESA_TAC azim[`x:real^3`;`u:real^3`;`w:real^3`;`z:real^3`]
THEN REMOVE_ASSUM_TAC
THEN MP_TAC(REAL_ARITH`&0<= azim x u w z ==> azim x u  w z = &0 \/ &0< azim x u w (z:real^3)`)
THEN RESA_TAC
THENL(*1*)[ MRESA_TAC AZIM_COMPL_EQ_0[`x:real^3`;`u:real^3`;`w:real^3`;`z:real^3`];(*1*)
ABBREV_TAC`w1=sigma_fan x V E u w:real^3`
THEN SUBGOAL_THEN`~(set_of_edge u V E = {w:real^3})`ASSUME_TAC
THENL(*2*)[
STRIP_TAC
THEN FIND_ASSUM (MP_TAC)`CARD (set_of_edge (u:real^3) V E) > 1`
THEN POP_ASSUM (fun th-> REWRITE_TAC[th])
THEN SUBGOAL_THEN `FINITE {(w:real^3)}` ASSUME_TAC
THENL (*3*)[SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY;
                 IN_INSERT; NOT_IN_EMPTY];
MRESAL_TAC CARD_PSUBSET[`{}:real^3->bool`;`{w:real^3}`][SET_RULE`{} PSUBSET {w}`]
THEN MRESAL_TAC CARD_DELETE[`w:real^3`;`{w:real^3}`][SET_RULE`w IN {w}/\ {w} DELETE w= {}`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[CARD_CLAUSES]
THEN ARITH_TAC];(*2*)

MRESA_TAC SIGMA_FAN[`x:real^3`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"LINH")
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w1:real^3`;`u:real^3`]
THEN POP_ASSUM MP_TAC
THEN RESA_TAC
THEN MRESA_TAC not_azim_points_in_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` U:real^3->bool`;`y:real^3`;`z:real^3`;`u:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th`w1:real^3`)
THEN REMOVE_THEN "YEU" (fun th-> MRESAL1_TAC th`w1:real^3`[azim1])
THEN MP_TAC(REAL_ARITH`&2 * pi - azim x u z w <= &2 * pi - azim x u z w1==> azim x u z (w1:real^3) <=  azim x u z w`)
THEN RESA_TAC
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`u:real^3`;`z:real^3`;`w1:real^3`;` w:real^3`]
THEN MP_TAC(REAL_ARITH` azim x u z w = azim x u z w1 + azim x u w1 w  /\ &0<= azim x u z w1
==> &2 * pi - azim x u z w <= &2 * pi - azim x u w1 (w:real^3)`)
THEN ASM_REWRITE_TAC[azim]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th)] THEN ASSUME_TAC(SYM(th)))
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"BE")
THEN SUBGOAL_THEN`~(azim x u w1 (w:real^3)= &0)` ASSUME_TAC
THENL[ STRIP_TAC
THEN MRESA_TAC UNIQUE_AZIM_0_POINT_FAN[`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(u:real^3)`;`(w1:real^3)`;`w:real^3`];

MRESA_TAC AZIM_COMPL[`x:real^3`;`u:real^3`;`z:real^3`;`w:real^3`]
THEN MRESA_TAC AZIM_COMPL[`x:real^3`;`u:real^3`;`w1:real^3`;`w:real^3`]
THEN STRIP_TAC
THEN MP_TAC(REAL_ARITH`&2 * pi - azim x u z w <= &2 * pi - azim x u w1 w
 ==> &2 * pi - azim x u z w < &2 * pi - azim x u w1 w \/  azim x u z w = azim x u w1 (w:real^3)`)
THEN RESA_TAC
THEN REMOVE_THEN"BE" MP_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`A+B=B<=>A= &0`]]]]);;




let aff_gt_in_w_dart_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) u:real^3 w:real^3 y:real^3.
FAN(x,V,E) /\ {u,w} IN E
/\ y IN w_dart_fan x V E ((x:real^3),(u:real^3),(w:real^3),(sigma_fan x V E u w:real^3)) 
/\ fan80(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
==> aff_gt {x} {u,y} SUBSET w_dart_fan x V E ((x:real^3),(u:real^3),(w:real^3),(sigma_fan x V E u w:real^3)) `,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "CON")
THEN DISCH_THEN(LABEL_TAC "BE")
THEN DISCH_THEN(LABEL_TAC"EM")
THEN USE_THEN "BE" MP_TAC
THEN REWRITE_TAC[fan80]
THEN DISCH_TAC
THEN POP_ASSUM (fun th->   MRESA_TAC th [`u:real^3`;`w:real^3`]
THEN MRESA_TAC th [`v:real^3`;`u:real^3`])
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`]
THEN REMOVE_THEN "EM" (fun th-> MRESA1_TAC th `u:real^3` THEN MP_TAC (th) THEN DISCH_THEN(LABEL_TAC"EM"))
THEN DISJ_CASES_TAC(SET_RULE`set_of_edge u V E = {w} \/ ~(set_of_edge u V E = {w:real^3})`)
THENL[
MRESA_TAC CARD_SING[`w:real^3`; `(set_of_edge u V E):real^3->bool`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th->REWRITE_TAC[SYM(th)])
THEN REMOVE_ASSUM_TAC THEN POP_ASSUM MP_TAC THEN ARITH_TAC;
REMOVE_THEN "CON" MP_TAC
THEN ASM_REWRITE_TAC[SUBSET; w_dart_fan;wedge;IN_ELIM_THM;]
THEN STRIP_TAC THEN GEN_TAC THEN STRIP_TAC
THEN MRESA_TAC aff_gt_inter_aff_gt [`(x:real^3)`;`(u:real^3)`;`(y:real^3)`]
THEN MP_TAC(SET_RULE`aff_gt {x} {u, y} = aff_gt {x, u} {y} INTER aff_gt {x, y} {u}
/\ x' IN aff_gt {x} {u, y:real^3} ==> x' IN aff_gt {x,u} {y}`) THEN RESA_TAC
THEN MRESA_TAC aff_gt_imp_not_collinear[`x:real^3`;`y:real^3`;`u:real^3`; `x':real^3`]
THEN MRESA_TAC AZIM_EQ_ALT[`x:real^3`;`u:real^3`;`w:real^3`;`x':real^3`;`y:real^3`;]]);;

let not_empty_rcone_fan_inter_aff_gt=prove(`!x v u:real^3 h:real.
~collinear {x,v,u} /\ &0< h /\ h<= pi==>
~(rcone_fan x v (cos h) INTER aff_gt {x} {v, u}={})`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN SUBGOAL_THEN`~(norm((v:real^3)-(x:real^3))= &0)` ASSUME_TAC
THENL(*1*)[ASM_REWRITE_TAC[NORM_EQ_0;VECTOR_ARITH`v-x=vec 0<=> x=v`];
ASM_REWRITE_TAC[rcone_fan;SET_RULE`~(A={})<=> ?y. y IN A`;INTER;IN_ELIM_THM]
THEN DISJ_CASES_TAC(REAL_ARITH`(v - x) dot (u - x:real^3) <= &0 \/ &0< (v - x) dot (u - x)`)
THENL(*2*)[ABBREV_TAC`s1= min h (pi / &2) / &2:real`
THEN MP_TAC(REAL_ARITH` &0< pi /\ min h (pi / &2) / &2 =s1 /\ &0< h:real ==> &0<= s1 /\ &0<s1 /\ s1< h/\ s1<pi/ &2`)
THEN ASM_REWRITE_TAC[PI_WORKS]
THEN STRIP_TAC
THEN EXISTS_TAC`sin (s1) % (e1_fan x v u) + (cos s1) %(e3_fan x v u)+x :real^3 `
THEN REWRITE_TAC[dist;vector_norm;VECTOR_ARITH`(B+C+A)-A=(B+C:real^3)`; DOT_LADD;DOT_RADD;DOT_RMUL;DOT_LMUL;]
THEN MRESAL_TAC properties_coordinate[`x:real^3`;`v:real^3`;`u:real^3`][orthonormal]
THEN ONCE_REWRITE_TAC[DOT_SYM]
THEN ASSUME_TAC(ISPEC`s1:real`SIN_CIRCLE)
THEN ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`sin s * sin s + cos s * cos s =(sin s) pow 2 + (cos s) pow 2`;e3_fan;DOT_RMUL; ]
THEN ONCE_REWRITE_TAC[GSYM vector_norm;]
THEN REWRITE_TAC[DOT_SQUARE_NORM;]
THEN MRESAL1_TAC  SQRT_POW_2`&1`[REAL_ARITH`a pow 2 = a * a`;REAL_ARITH`&0<= &1`;]
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC SQRT_MUL[`&1`;`&1`][REAL_ARITH`&0<= &1`;]
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th); REAL_ARITH`&1* &1= &1`])
THEN RESA_TAC
THEN REDUCE_ARITH_TAC
THEN MP_TAC(ISPEC`norm((v:real^3)-(x:real^3))`REAL_MUL_LINV)
THEN REWRITE_TAC[REAL_ARITH`A*B*D *D=(B*D) *D *A`;REAL_ARITH`A>B<=> B<A`]
THEN RESA_TAC
THEN REDUCE_ARITH_TAC
THEN ASSUME_TAC(ISPEC`v-x:real^3`NORM_POS_LE)
THEN STRIP_TAC
THENL(*3*)[ MATCH_MP_TAC REAL_LT_LMUL
THEN MP_TAC(REAL_ARITH`~(norm (v - x:real^3) = &0)/\ &0 <= norm (v - x)==> &0< norm (v - x) `)
THEN RESA_TAC
THEN MATCH_MP_TAC COS_MONO_LT
THEN ASM_REWRITE_TAC[];(*3*)
MRESA_TAC condition1_to_in_aff_gt_by_angle[`x:real^3`;`v:real^3`;`u:real^3`;`s1:real`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[e3_fan]](*3*);(*2*)
SUBGOAL_THEN`&0<(atn ((norm ((v - x) cross (u - x))) * inv((v - x) dot (u - x:real^3))))` ASSUME_TAC
THENL(*3*)[MP_TAC(ISPEC`(v - x) dot (u - x:real^3)`REAL_LT_INV)
THEN RESA_TAC
THEN ASSUME_TAC(PI_WORKS)
THEN MP_TAC(REAL_ARITH`&0< pi ==> --(pi / &2) < &0`)
THEN RESA_TAC
THEN MRESAL_TAC  ATN_MONO_LT[`&0:real`;` (norm ((v - x) cross (u - x)) * inv ((v - x) dot (u - x))):real`][ ATN_0]
THEN POP_ASSUM MATCH_MP_TAC
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN SUBGOAL_THEN`~(norm((v - x) cross (u - x:real^3))= &0)` ASSUME_TAC
THENL(*4*)[
ASM_REWRITE_TAC[NORM_EQ_0]
THEN MP_TAC(ISPECL[`v-x:real^3`;`u-x:real^3`]CROSS_EQ_0)
THEN ONCE_REWRITE_TAC[GSYM COLLINEAR_3;]
THEN RESA_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN ASM_REWRITE_TAC[];(*4*)

POP_ASSUM MP_TAC
THEN MP_TAC(ISPEC`(v - x) cross (u - x:real^3)`NORM_POS_LE)
THEN REAL_ARITH_TAC](*4*);(*3*)

 ASSUME_TAC(ISPEC`(norm ((v - x) cross (u - x)) * inv ((v - x) dot (u - x))):real`ATN_BOUNDS)
THEN ABBREV_TAC`s2= atn ((norm ((v - x) cross (u - x))) * inv((v - x) dot (u - x))):real`
THEN ABBREV_TAC`s1= min (h:real) (s2:real) / &2`
THEN MP_TAC(REAL_ARITH`&0< h /\ s1= min (h:real) (s2) / &2  /\ &0< pi /\  &0< s2 /\ s2 < pi/ &2==> &0<= s1 /\ &0<s1 /\ s1<pi/ &2 /\ s1<h/\  s1< s2
`)
THEN REWRITE_TAC[PI_WORKS]
THEN RESA_TAC
THEN EXISTS_TAC`sin (s1) % (e1_fan x v u) + (cos s1) %(e3_fan x v u)+x :real^3 `
THEN REWRITE_TAC[dist;vector_norm;VECTOR_ARITH`(B+C+A)-A=(B+C:real^3)`; DOT_LADD;DOT_RADD;DOT_RMUL;DOT_LMUL;]
THEN MRESAL_TAC properties_coordinate[`x:real^3`;`v:real^3`;`u:real^3`][orthonormal]
THEN ONCE_REWRITE_TAC[DOT_SYM]
THEN ASSUME_TAC(ISPEC`s1:real`SIN_CIRCLE)
THEN ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`sin s * sin s + cos s * cos s =(sin s) pow 2 + (cos s) pow 2`;e3_fan;DOT_RMUL; ]
THEN ONCE_REWRITE_TAC[GSYM vector_norm;]
THEN REWRITE_TAC[DOT_SQUARE_NORM;]
THEN MRESAL1_TAC  SQRT_POW_2`&1`[REAL_ARITH`a pow 2 = a * a`;REAL_ARITH`&0<= &1`;]
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC SQRT_MUL[`&1`;`&1`][REAL_ARITH`&0<= &1`;]
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th); REAL_ARITH`&1* &1= &1`])
THEN RESA_TAC
THEN REDUCE_ARITH_TAC
THEN MP_TAC(ISPEC`norm((v:real^3)-(x:real^3))`REAL_MUL_LINV)
THEN REWRITE_TAC[REAL_ARITH`A*B*D *D=(B*D) *D *A`;REAL_ARITH`A>B<=> B<A`]
THEN RESA_TAC
THEN REDUCE_ARITH_TAC
THEN ASSUME_TAC(ISPEC`v-x:real^3`NORM_POS_LE)
THEN STRIP_TAC
THENL(*4*)[ MATCH_MP_TAC REAL_LT_LMUL
THEN MP_TAC(REAL_ARITH`~(norm (v - x:real^3) = &0)/\ &0 <= norm (v - x)==> &0< norm (v - x) `)
THEN RESA_TAC
THEN MATCH_MP_TAC COS_MONO_LT
THEN ASM_REWRITE_TAC[];(*4*)
MRESA_TAC condition_to_in_aff_gt_by_angle[`x:real^3`;`v:real^3`;`u:real^3`;`s1:real`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[e3_fan]]]]]);;





let condition_rw_dart_fan_inter_aff_gt_is_not_empty=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 z:real^3 h:real.
FAN(x,V,E)/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E) /\ ~collinear {x,v,z}
/\ {v,u} IN E /\ z IN w_dart_fan x V E (x,v,u,sigma_fan x V E v u)
/\ &0<h /\ h<= pi
==>
~(rw_dart_fan x V E ((x:real^3),(v:real^3),(u:real^3),sigma_fan x V E v u ) (cos(h)) INTER aff_gt {x} {v,z}={})`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC aff_gt_in_w_dart_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;` u:real^3`;`z:real^3`]
THEN MP_TAC(SET_RULE`aff_gt {x} {v, z} SUBSET w_dart_fan x V E (x,v,u,sigma_fan x V E v u) ==>
(w_dart_fan x V E (x,v,u,sigma_fan x V E v u) INTER
      rcone_fan x v (cos h)) INTER
     aff_gt {x} {v, z}=  rcone_fan x v (cos h) INTER
     aff_gt {x} {v, z:real^3}`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[rw_dart_fan;SET_RULE`~(A={})<=> ?y. y IN A`]
THEN MRESA_TAC not_empty_rcone_fan_inter_aff_gt[`x:real^3`;`v:real^3`;`z:real^3`;`h:real`]
THEN POP_ASSUM MP_TAC
THEN SET_TAC[]);;



let exists_edge_rw_dart_fan_inter_aff_gt_not_empty_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool y:real^3 z:real^3 u:real^3.
FAN(x,V,E) 
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ U IN topological_component_yfan (x,V,E)
/\ z IN U
/\ u IN V
/\ y IN aff_ge {x} {u}
/\ y IN xfan(x,V,E)
/\ ~(y=x)
/\(!t. &0< t /\ t< &1==>   (&1-t)%y+t%z IN yfan (x,V,E))
==> ?w. {u,w} IN E /\
(!h. &0<h /\ h<= pi
==> ~((rw_dart_fan x V E (x,u,w,sigma_fan x V E u w) (cos h)) INTER aff_gt {x} {u,z}={}))`,

REPEAT STRIP_TAC
THEN MRESA_TAC exists_edge_bounded_topological_component_yfan[`x:real^3`;`(V:real^3->bool)`; `(E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;`z:real^3`;`u:real^3`]
THEN EXISTS_TAC`w:real^3`
THEN ASM_REWRITE_TAC[]
THEN GEN_TAC THEN STRIP_TAC
THEN MATCH_MP_TAC condition_rw_dart_fan_inter_aff_gt_is_not_empty
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`u:real^3`]
THEN MRESA_TAC point_in_yfan_and_point_in_xfan_indepent_fan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;` z:real^3`]
THEN MRESA_TAC aff_ge_1_1_subset_aff_fan[`y:real^3`;`x:real^3`;`u:real^3`]
THEN MRESA_TAC permutes_4points_collinear[`x:real^3`;`y:real^3`;`u:real^3`;`z:real^3`]);;




let exists_edge_rw_dart_fan_inter_topological_component_not_empty_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool y:real^3 z:real^3 u:real^3.
FAN(x,V,E) 
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ U IN topological_component_yfan (x,V,E)
/\ z IN U
/\ u IN V
/\ y IN aff_ge {x} {u}
/\ y IN xfan(x,V,E)
/\ ~(y=x)
/\(!t. &0< t /\ t< &1==>   (&1-t)%y+t%z IN yfan (x,V,E))

==> ?w. {u,w} IN E 
/\(!h.  &0<h /\ h<= pi
==> ~((rw_dart_fan x V E (x,u,w,sigma_fan x V E u w) (cos h)) INTER U={}))`,

REPEAT STRIP_TAC
THEN MRESA_TAC nonsetedge_fully_surround_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`]
THEN MRESA_TAC point_in_yfan_not_x_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;` z:real^3`]
THEN MP_TAC(SET_RULE`~(x=z) /\ ~(y=x)==> DISJOINT {x} {y,z:real^3}`)
THEN RESA_TAC
THEN MRESA_TAC aff_gt_subset_component_y_fan [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;`z:real^3`]
THEN MRESA_TAC exists_edge_rw_dart_fan_inter_aff_gt_not_empty_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;`z:real^3`;`u:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"YEU")
THEN EXISTS_TAC `w:real^3`
THEN ASM_REWRITE_TAC[]
THEN GEN_TAC THEN STRIP_TAC
THEN REMOVE_THEN "YEU"(fun th-> MRESA1_TAC th`h:real`)
THEN MATCH_MP_TAC(SET_RULE`~(rw_dart_fan x V E (x,u,w,sigma_fan x V E u w) (cos h) INTER aff_gt {x} {u, z} =  {})
/\ aff_gt {x:real^3} {y, z} SUBSET U /\ aff_gt {x} {u, z} =aff_gt {x} {y, z} 
==> ~(rw_dart_fan x V E (x,u,w,sigma_fan x V E u w) (cos h) INTER U = {})`)
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC aff_gt_1_2_scale_fan
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` w:real^3`;`u:real^3`]
THEN MRESA_TAC aff_ge_1_1_subset_aff_fan[`y:real^3`;`x:real^3`;`u:real^3`]
THEN MRESA_TAC point_in_yfan_and_point_in_xfan_indepent_fan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;` z:real^3`]
THEN MRESA_TAC permutes_4points_collinear[`x:real^3`;`y:real^3`;`u:real^3`;`z:real^3`]
THEN ASM_REWRITE_TAC[]
THEN FIND_ASSUM MP_TAC `y IN aff_ge {x} {u:real^3}`
THEN FIND_ASSUM MP_TAC ` ~(x=u:real^3)`
THEN FIND_ASSUM MP_TAC ` ~(y=x:real^3)`
THEN REPEAT REMOVE_ASSUM_TAC
THEN STRIP_TAC
THEN STRIP_TAC
THEN MRESAL_TAC AFF_GE_1_1[`x:real^3`;`u:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`t2:real`
THEN ASM_REWRITE_TAC[VECTOR_ARITH` (t1 % x + t2 % u) - x= ((t1+t2)- &1) %x+ t2%(u-x)`;]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(REAL_ARITH`&0<= t2==> t2= &0 \/ &0< t2`)
THEN RESA_TAC
THEN REDUCE_ARITH_TAC
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]
THEN VECTOR_ARITH_TAC);;





let  exists_dart_leads_into_edge_eq_topological_component_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool y:real^3 z:real^3 u:real^3.
FAN(x,V,E) 
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ U IN topological_component_yfan (x,V,E)
/\ z IN U
/\ u IN V
/\ y IN aff_ge {x} {u}
/\ y IN xfan(x,V,E)
/\ ~(y=x)
/\(!t. &0< t /\ t< &1==>   (&1-t)%y+t%z IN yfan(x,V,E))
==> ?w. {u,w} IN E /\ dart_leads_into x V E u w = U`,

REPEAT STRIP_TAC
THEN MRESA_TAC exists_edge_rw_dart_fan_inter_topological_component_not_empty_fan[`x:real^3`;`(V:real^3->bool)`; `(E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;`z:real^3`;`u:real^3`;]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"MA")
THEN MRESA_TAC not_empty_rw_dart_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"EM")
THEN MRESA_TAC DART_LEADS_INTO[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;
`u:real^3`;`w:real^3`;]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"YEU")
THEN MRESA_TAC rw_dart_avoids_fan[`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"OI")
THEN MP_TAC(REAL_ARITH`&1> h' /\ h' > &0==> -- &1 < h' /\ h'< &1 /\ -- &1 <= h' /\ h'<= &1/\ &0 < h' /\ h' <= &1`) THEN RESA_TAC
THEN MRESA1_TAC ACS_BOUNDS_LT`h':real`
THEN MRESAL_TAC ACS_MONO_LT[`&0`;`h':real`][ACS_0;REAL_ARITH`-- &1 <= &0`]
THEN MRESA1_TAC COS_ACS `h':real`
THEN ABBREV_TAC`h1= min (h:real) (acs h')/ &2`
THEN MP_TAC(REAL_ARITH`h1= min (h:real) (acs h')/ &2 /\ &0<h /\ &0< acs h' /\ acs h'< pi/ &2==> &0< h1 /\ h1< h /\ h1<pi/ &2/\ h1< acs h' /\ acs h' <= pi /\ &0<= h1/\ h1<= pi`)
THEN ASM_REWRITE_TAC[PI_WORKS] THEN STRIP_TAC
THEN MRESAL_TAC COS_MONO_LT[`h1:real`;`acs h':real`][ACS_0;REAL_ARITH`-- &1 <= &0`]
THEN MP_TAC(REAL_ARITH`h'< cos h1==> h'<= cos h1`) THEN RESA_TAC
THEN REMOVE_THEN "EM"(fun th-> MRESAL1_TAC th `h1:real`[SET_RULE`~(A={})<=> ?y. y IN A`])
THEN REMOVE_THEN "YEU"(fun th-> MRESA_TAC th [`h1:real`;`y':real^3`])
THEN EXISTS_TAC `w:real^3`
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "EM")
THEN DISCH_TAC
THEN REMOVE_THEN "EM" MP_TAC
THEN POP_ASSUM(fun th -> REWRITE_TAC[SYM(th);]) 
THEN DISCH_TAC
THEN MRESA_TAC expand_element_in_topological_component_yfan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`U:real^3->bool`;`z:real^3`]
THEN REMOVE_THEN "MA"(fun th-> MRESA1_TAC th `h1:real`)
THEN MP_TAC(SET_RULE`rw_dart_fan x V E (x,u,w,sigma_fan x V E u w) (cos h1) SUBSET
      connected_component (yfan (x,V,E)) y'/\
~(rw_dart_fan x V E (x,u,w,sigma_fan x V E u w) (cos h1) INTER
        connected_component (yfan (x,V,E)) z =
        {})
==>
~(   connected_component (yfan (x,V,E)) y' INTER
        connected_component (yfan (x,V,E)) z =
        {})`)
THEN ASM_REWRITE_TAC[]
THEN REWRITE_TAC[CONNECTED_COMPONENT_OVERLAP]
THEN SET_TAC[]);;

(*CASE 2*)



let not_azim_points1_in_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool y:real^3 z:real^3 u:real^3 w:real^3.
FAN(x,V,E) 
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ U IN topological_component_yfan (x,V,E)
/\ z IN U
/\ {u,w} IN E
/\ y IN aff_gt {x} {u,w}
/\ y IN xfan (x,V,E)
/\ ~(y=x)
 /\(!t. &0< t /\ t< &1==>   (&1-t)%y+t%z IN yfan (x,V,E))
==>  ~(azim x y z w= &0)`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"YEU")
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`]
THEN MRESA_TAC nonsetedge_fully_surround_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`]
THEN MRESA_TAC point_in_yfan_not_x_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;` z:real^3`]
THEN MRESA_TAC point_in_yfan_and_point_in_xfan_indepent_fan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;` z:real^3`]
THEN MRESA_TAC properties_of_collinear4_points_fan[`x:real^3`;`w:real^3`;`u:real^3`;`y:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`;SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`;]
THEN STRIP_TAC
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`y:real^3`;`z:real^3`;`w:real^3`]
THEN STRIP_TAC
THEN MRESA_TAC th3[`x:real^3`;`y:real^3`;`w:real^3`]
THEN MRESA_TAC aff_ge_2_1_is_exists_point_inaff_ge_1_2[`x:real^3`;`y:real^3`;`z:real^3`;`w:real^3`]
THEN MRESA_TAC aff_ge_eq_aff_gt_union_aff_ge[`x:real^3`;`u:real^3`;`w:real^3`]
THEN MP_TAC(SET_RULE`aff_ge {x} {u, w} =
      aff_gt {x} {u, w} UNION aff_ge {x} {u} UNION aff_ge {x} {w}
/\ y IN aff_gt {x} {u,w}==> y IN aff_ge {x:real^3} {u,w}`)
THEN POP_ASSUM(fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN RESA_TAC
THEN MRESA_TAC aff_ge1_subset_aff_ge[`x:real^3`;`u:real^3`; `w:real^3`;`y:real^3`]
THEN SUBGOAL_THEN`aff_ge {x} {u,w} SUBSET xfan(x:real^3,V:real^3->bool,E)` ASSUME_TAC
THENL[
ASM_REWRITE_TAC[xfan; SUBSET;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`{u,w:real^3}`
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN SET_TAC[IN];
MP_TAC(SET_RULE`(&1 - t) % y + t % z IN aff_ge {x:real^3} {y, w}/\ aff_ge {x} {y, w} SUBSET aff_ge {x} {u, w}  /\aff_ge {x} {u, w} SUBSET xfan (x,V:real^3->bool,E)
==> (&1 - t) % y + t % z IN  xfan (x,V,E)`)
THEN RESA_TAC
THEN REMOVE_THEN "YEU"(fun th -> MRESA1_TAC th`t:real` )
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[yfan]
THEN SET_TAC[]]);;


let not_azim_points2_in_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool y:real^3 z:real^3 u:real^3 w:real^3.
FAN(x,V,E) 
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ U IN topological_component_yfan (x,V,E)
/\ z IN U
/\ {u,w} IN E
/\ y IN aff_gt {x} {u,w}
/\ y IN xfan (x,V,E)
/\ ~(y=x)
 /\(!t. &0< t /\ t< &1==>   (&1-t)%y+t%z IN yfan (x,V,E))
==>  ~(azim x y z u= &0)`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[]
THEN MATCH_MP_TAC not_azim_points1_in_yfan
THEN EXISTS_TAC`V:real^3->bool`
THEN EXISTS_TAC`E:(real^3->bool)->bool`
THEN EXISTS_TAC`U:real^3->bool`
THEN EXISTS_TAC`w:real^3`
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]);;





let condition_cross_dot_4point=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
              let va = a1 cross a2 in
              let vb = a3 cross a4 in
              let v3 = va cross vb + x in 
 ~collinear {x,v,u}
/\ &0<(a1 cross a2) dot a4 /\  &0 < --((a1 cross a2) dot a3)
==> v3 IN aff_gt {x} {v,u}`,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN ABBREV_TAC`a1=(y-x):real^3`
THEN ABBREV_TAC`a2=(z-x):real^3`
THEN ABBREV_TAC`a3=(v-x) :real^3`
THEN ABBREV_TAC`a4=u-x:real^3`
THEN ABBREV_TAC`va=a1 cross a2:real^3`
THEN ABBREV_TAC`vb=a3 cross a4:real^3`
THEN ABBREV_TAC`v3= (va:real^3) cross (vb:real^3)+(x:real^3)`
THEN REPEAT STRIP_TAC
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(u:real^3) `;]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&1-(va:real^3) dot (a4:real^3)+va dot (a3:real^3)`
THEN EXISTS_TAC`((va:real^3) dot (a4:real^3))`
THEN EXISTS_TAC`(--((va:real^3) dot (a3:real^3)))`
THEN ASM_REWRITE_TAC[REAL_ARITH` (&1 - va dot a4 + va dot a3) + (va dot a4) + --(va dot a3) = &1`]
THEN EXPAND_TAC"v3"
THEN EXPAND_TAC"vb" 
THEN REWRITE_TAC[CROSS_LAGRANGE;VECTOR_ARITH`A+ B + --U%C=A +B-U%C:real^3`]
THEN EXPAND_TAC"a3" 
THEN EXPAND_TAC"a4" 
THEN VECTOR_ARITH_TAC);;



let aff_gt_2_1_crossr_dot_4point=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 ~collinear {x,y,z}
/\ u IN aff_gt {x,y} {z}
/\ &0<(a1 cross a2) dot a3 
==> &0<(a1 cross a4) dot a3 `,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`y:real^3`;`z:real^3`]
THEN MRESAL_TAC AFF_GT_2_1[`x:real^3`;`y:real^3`;`z:real^3`][IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(u' % x + v' % y + w % z) - x=((u'+v'+w) - &1) % x + v' % (y-x) + w % (z - x)`;REAL_ARITH`&1- &1= &0`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[CROSS_RMUL;CROSS_RADD;CROSS_REFL;]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[DOT_LMUL]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]);;

let aff_gt_2_1_rcross_dot_4pointl=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 ~collinear {x,y,z}
/\ u IN aff_gt {x,z} {y}
/\ &0<(a1 cross a2) dot a3 
==> &0<(a4 cross a2) dot a3 `,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`z:real^3`;`y:real^3`]
THEN MRESAL_TAC AFF_GT_2_1[`x:real^3`;`z:real^3`;`y:real^3`][IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(u' % x + v' % z + w % y) - x=((u'+v'+w) - &1) % x + v' % (z-x) + w % (y - x)`;REAL_ARITH`&1- &1= &0`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[CROSS_LMUL;CROSS_LADD;CROSS_REFL;]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[DOT_LMUL]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]);;



let aff_gt_2_1_cross_dotr_4point=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 ~collinear {x,y,v}
/\ u IN aff_gt {x,y} {v}
/\ &0<(a1 cross a2) dot a3 
==> &0<(a1 cross a2) dot a4 `,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`y:real^3`;`v:real^3`]
THEN MRESAL_TAC AFF_GT_2_1[`x:real^3`;`y:real^3`;`v:real^3`][IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(u' % x + v' % z + w % y) - x=((u'+v'+w) - &1) % x + v' % (z-x) + w % (y - x)`;REAL_ARITH`&1- &1= &0`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[DOT_RMUL;DOT_RADD;DOT_CROSS_SELF]
THEN REDUCE_ARITH_TAC
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]);;

let aff_gt_2_1_cross_dotl_4point=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 ~collinear {x,z,v}
/\ u IN aff_gt {x,z} {v}
/\ &0<(a1 cross a2) dot a3 
==> &0<(a1 cross a2) dot a4 `,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`z:real^3`;`v:real^3`]
THEN MRESAL_TAC AFF_GT_2_1[`x:real^3`;`z:real^3`;`v:real^3`][IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(u' % x + v' % z + w % y) - x=((u'+v'+w) - &1) % x + v' % (z-x) + w % (y - x)`;REAL_ARITH`&1- &1= &0`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[DOT_RMUL;DOT_RADD;DOT_CROSS_SELF]
THEN REDUCE_ARITH_TAC
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]);;




let aff_gt_2_1r_rcross_dotl_4point=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 ~collinear {x,v,y}
/\ u IN aff_gt {x,v} {y}
/\ &0<(a1 cross a2) dot a3 
==> &0<(a4 cross a2) dot a3 `,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`y:real^3`]
THEN MRESAL_TAC AFF_GT_2_1[`x:real^3`;`v:real^3`;`y:real^3`][IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(u' % x + v' % z + w % y) - x=((u'+v'+w) - &1) % x + v' % (z-x) + w % (y - x)`;REAL_ARITH`&1- &1= &0`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[CROSS_LMUL;CROSS_LADD;DOT_LMUL;DOT_LADD;DOT_CROSS_SELF]
THEN REDUCE_ARITH_TAC
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]);;




let aff_gt_1_2_cross_dotr_4point=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 ~collinear {x,v,u}
/\ y IN aff_gt {x} {v,u}
/\ &0<(a1 cross a2) dot a3
 ==> &0< --((a1 cross a2) dot a4)`,

REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESAL_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(u' % x + v' % z + w % y) - x=((u'+v'+w) - &1) % x + v' % (z-x) + w % (y - x)`;REAL_ARITH`&1- &1= &0`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[CROSS_LNEG;CROSS_LMUL;CROSS_LADD;CROSS_REFL;DOT_LMUL;DOT_LADD;]
THEN ONCE_REWRITE_TAC[CROSS_SKEW;CROSS_TRIPLE]
THEN REWRITE_TAC[CROSS_TRIPLE;CROSS_REFL;DOT_LZERO]
THEN REDUCE_ARITH_TAC
THEN STRIP_TAC
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[DOT_LNEG;REAL_ARITH`--(A* (--B))=A*B`]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE;]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC REAL_LT_RCANCEL_IMP[`&0`;`((u - x) cross (z - x)) dot (v - x:real^3)`;`t3:real`;][REAL_ARITH`&0 * A= &0`]
THEN POP_ASSUM MATCH_MP_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC);;




let aff_gt_1_2_cross_dotr_4point_neg=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 ~collinear {x,v,u}
/\ y IN aff_gt {x} {v,u}
/\ &0< --((a1 cross a2) dot a3)
 ==> &0< ((a1 cross a2) dot a4)`,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESAL_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(u' % x + v' % z + w % y) - x=((u'+v'+w) - &1) % x + v' % (z-x) + w % (y - x)`;REAL_ARITH`&1- &1= &0`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[CROSS_LNEG;CROSS_LMUL;CROSS_LADD;CROSS_REFL;DOT_LMUL;DOT_LADD;]
THEN ONCE_REWRITE_TAC[CROSS_SKEW;CROSS_TRIPLE]
THEN REWRITE_TAC[CROSS_TRIPLE;CROSS_REFL;DOT_LZERO]
THEN REDUCE_ARITH_TAC
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[DOT_LNEG;REAL_ARITH`--(A* (--B))=A*B`]
THEN STRIP_TAC
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[DOT_LNEG;REAL_ARITH`-- --A=A`]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE;]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE;]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC REAL_LT_RCANCEL_IMP[`&0`;`((z - x) cross (u - x)) dot (v - x):real^3`;`t3:real`;][REAL_ARITH`&0 * A= &0`]
THEN POP_ASSUM MATCH_MP_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC);;


let aff_gt_1_2_cross_dotr_4point_zero=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 ~collinear {x,v,u}
/\ y IN aff_gt {x} {v,u}
/\ (a1 cross a2) dot a3= &0
 ==> ((a1 cross a2) dot a4)= &0`,

REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESAL_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(u' % x + v' % z + w % y) - x=((u'+v'+w) - &1) % x + v' % (z-x) + w % (y - x)`;REAL_ARITH`&1- &1= &0`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[CROSS_LNEG;CROSS_LMUL;CROSS_LADD;CROSS_REFL;DOT_LMUL;DOT_LADD;]
THEN ONCE_REWRITE_TAC[CROSS_SKEW;CROSS_TRIPLE]
THEN REWRITE_TAC[CROSS_TRIPLE;CROSS_REFL;DOT_LZERO]
THEN REDUCE_ARITH_TAC
THEN STRIP_TAC
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[DOT_LNEG;REAL_ARITH`--(A* (--B))=A*B`]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE;]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[REAL_ENTIRE]
THEN STRIP_TAC
THEN ASM_TAC THEN REAL_ARITH_TAC);;

let exists_esilon_real=prove(`!a:real b:real.
&0<a ==> ?t. &0< t /\ t< &1 /\ 
(!h. &0< h /\ h< t==> &0< a- h * b)`,
REPEAT STRIP_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`b <= &0 \/ &0< b`)
THENL[ EXISTS_TAC`&1/ &2`
THEN REWRITE_TAC[REAL_ARITH`&0< &1/ &2 /\ &1/ &2< &1`;]
THEN REPEAT STRIP_TAC
THEN MATCH_MP_TAC(REAL_ARITH`&0<a /\ &0<= h*(-- b)==> &0< a-h*b`)
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= --B<=> B<= &0`]
THEN ASM_TAC THEN REAL_ARITH_TAC;
ABBREV_TAC`t1= (min (inv (b:real) * a) (&1)) / &2`
THEN MRESA1_TAC REAL_LT_INV`b:real`
THEN MRESA_TAC REAL_LT_MUL[`inv b:real`;`a:real`]
THEN MP_TAC(REAL_ARITH`&0 < inv b * a /\ t1= (min (inv (b:real) * a) (&1)) / &2
==> &0< t1 /\ t1< &1 /\ t1< inv b * a`)
THEN RESA_TAC
THEN EXISTS_TAC `t1:real`
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`h<t1 /\ t1< inv b *a==> &0< inv b *a- h`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`b:real`;`inv b *a- h:real`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`b * (inv b * a - h)= (inv b * b) * a- h *b `]
THEN MP_TAC(REAL_ARITH`&0<b==> ~(b= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`b:real`
THEN REAL_ARITH_TAC]);;


let invariant_cross_dotr_esilon_3piont=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 &0<(a1 cross a2) dot a3
 ==>
?t. &0< t /\ t< &1 /\ 
(!h. &0< h /\ h< t==>
 &0< ((a1 cross a2) dot ((&1 - h) % v + h % u-x)))`,

REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - h) % v + h % u-x=(&1 - h) % (v-x) + h % (u-x)`;]
THEN REWRITE_TAC[DOT_RMUL;DOT_RADD;]
THEN REWRITE_TAC[REAL_ARITH`(&1-h)*A+h*B=A-h*(A-B)`]
THEN MATCH_MP_TAC exists_esilon_real
THEN ASM_REWRITE_TAC[]);;


let invariant_rcross_dot_esilon_3piont=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 &0<(a1 cross a2) dot a3
 ==>
?t. &0< t /\ t< &1 /\ 
(!h. &0< h /\ h< t==>
 &0< (((&1 - h) % y + h % u-x) cross a2) dot a3)`,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - h) % v + h % u-x=(&1 - h) % (v-x) + h % (u-x)`;]
THEN REWRITE_TAC[CROSS_LMUL;CROSS_LADD;DOT_LMUL;DOT_LADD;]
THEN REWRITE_TAC[REAL_ARITH`(&1-h)*A+h*B=A-h*(A-B)`]
THEN MATCH_MP_TAC exists_esilon_real
THEN ASM_REWRITE_TAC[]);;

let invariant_crossr_dot_esilon_3piont=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 &0<(a1 cross a2) dot a3
 ==>
?t. &0< t /\ t< &1 /\ 
(!h. &0< h /\ h< t==>
 &0< (a1 cross ((&1 - h) % z + h % u-x)) dot a3)`,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - h) % v + h % u-x=(&1 - h) % (v-x) + h % (u-x)`;]
THEN REWRITE_TAC[CROSS_RMUL;CROSS_RADD;DOT_LMUL;DOT_LADD;]
THEN REWRITE_TAC[REAL_ARITH`(&1-h)*A+h*B=A-h*(A-B)`]
THEN MATCH_MP_TAC exists_esilon_real
THEN ASM_REWRITE_TAC[]);;

let point_in_aff_gt_2_1_change_point_in_aff_gt_1_2=prove(` !x:real^3 v:real^3 u:real^3 y:real^3.
 ~collinear {x,v,u}
/\  y IN aff_gt {x} {v,u}
==> u IN aff_gt {x,v} {y}`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"YEU")
THEN MRESA_TAC properties_of_collinear4_points_fan[`x:real^3`;`v:real^3`;`u:real^3`;`y:real^3`]
THEN REMOVE_THEN "YEU" MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`;]
THEN MRESA_TAC th3[`x:real^3`;`y:real^3`;`v:real^3`;]
THEN MRESA_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`;]
THEN MRESAL_TAC AFF_GT_2_1[`x:real^3`;`v:real^3`;`y:real^3`;][IN_ELIM_THM]
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0< t3==> ~(t3= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`t3:real`
THEN MRESA1_TAC REAL_MUL_LINV`t3:real`
THEN EXISTS_TAC`-- inv t3 * t1:real`
THEN EXISTS_TAC`-- inv t3 * t2:real`
THEN EXISTS_TAC`inv t3 :real`
THEN ASM_REWRITE_TAC[REAL_ARITH`--inv t3 * t1 + --inv t3 * t2 + inv t3= inv t3 *( t3 + &1- (t1+t2+t3))`;REAL_ARITH`A+ &1- &1 =A`;VECTOR_ARITH`(--inv t3 * t1) % x +
 (--inv t3 * t2) % v +
 inv t3 % (t1 % x + t2 % v + t3 % u)= (inv t3 * t3) % u`]
THEN VECTOR_ARITH_TAC);;
let pos_in_aff_gt_fan=prove(`!x:real^3 v:real^3 u:real^3 a:real.
DISJOINT {x} {v,u}
/\ &0<a /\ a< &1
==>
(&1-a)%v + a % u IN aff_gt {x} {v,u:real^3}`,

REPEAT STRIP_TAC
THEN MRESAL_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1 -a:real`
THEN EXISTS_TAC`a:real`
THEN MP_TAC(REAL_ARITH`&0< a /\ a  < &1 ==> &0 < &1 - a /\ &0 < a`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC);;

let pos_in_aff_gt_2_1_fan=prove(`!x:real^3 v:real^3 u:real^3 a:real.
DISJOINT {x,v} {u}
/\ &0<a /\ a< &1
==>
(&1-a)%v + a % u IN aff_gt {x,v} {u:real^3}`,

REPEAT STRIP_TAC
THEN MRESAL_TAC AFF_GT_2_1[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1 -a:real`
THEN EXISTS_TAC`a:real`
THEN MP_TAC(REAL_ARITH`&0< a /\ a  < &1 ==> &0 < &1 - a /\ &0 < a`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC);;


let condition_4point_aff_gt_1_2inter_aff_gt_1_2=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3 w:real^3 a:real.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
              let a5 = w - x in

 ~collinear {x,v,u}
/\ ~collinear {x,u,w}
/\ ~collinear {x,y,z}
/\  &0< a /\ a< &1 
/\ y IN aff_gt {x} {v,u}
/\ &0<(a3 cross a4) dot a5
/\ (!h. &0< h /\ h< a==> ~collinear {x,v,(&1-h)%u+h%w})
/\ &0<(a3 cross a1) dot a2
 ==> ?t. &0< t /\ t< &1 /\ 
(!h. &0< h /\ h< t==> ~(aff_gt {x} {y,z} INTER aff_gt {x} {v,(&1-h)%u+h%w}={}))`,

REPEAT STRIP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"LINH")
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN STRIP_TAC 
THEN MRESA_TAC aff_gt_1_2_cross_dotr_4point[`x:real^3`;`y:real^3`;`z:real^3`;`v:real^3`;`u:real^3`;]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN RESA_TAC
THEN MRESA_TAC invariant_cross_dotr_esilon_3piont[`x:real^3`; `z:real^3`;`y:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN ONCE_REWRITE_TAC[CROSS_SKEW]
THEN ASM_REWRITE_TAC[DOT_LNEG]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"YEU")
THEN MRESA_TAC properties_of_collinear4_points_fan[`x:real^3`;`v:real^3`;`u:real^3`;`y:real^3`]
THEN MRESA_TAC point_in_aff_gt_2_1_change_point_in_aff_gt_1_2[`x:real^3`;`v:real^3`;`u:real^3`;`y:real^3`]
THEN MRESA_TAC aff_gt_2_1r_rcross_dotl_4point[`x:real^3`;`y:real^3`;`z:real^3`;`v:real^3`;`u:real^3`]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN RESA_TAC
THEN MRESA_TAC invariant_rcross_dot_esilon_3piont[`x:real^3`; `u:real^3`;`z:real^3`;`v:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"EM")
THEN ABBREV_TAC`t1= min (min t t') a:real`
THEN MP_TAC(REAL_ARITH`&0<t/\ t< &1 /\ &0< t' /\ t'< &1 /\ &0< a /\ a < &1 /\ t1=min (min t t') a ==> &0< t1 /\ t1 < &1`)
THEN RESA_TAC
THEN EXISTS_TAC`t1:real`
THEN ASM_REWRITE_TAC[SET_RULE`~(A={})<=> ?y1. y1 IN A`]
THEN REPEAT STRIP_TAC
THEN ABBREV_TAC`a1=(y-x):real^3`
THEN ABBREV_TAC`a2=(z-x):real^3`
THEN ABBREV_TAC`a3=(v-x) :real^3`
THEN ABBREV_TAC`a4=(&1 - h) % u + h % w-x:real^3`
THEN ABBREV_TAC`va=a1 cross a2:real^3`
THEN ABBREV_TAC`vb=a3 cross a4:real^3`
THEN ABBREV_TAC`v3= (vb:real^3) cross (va:real^3)+(x:real^3)`
THEN EXISTS_TAC `v3:real^3`
THEN MP_TAC(REAL_ARITH`h<t1 /\ t1< &1 /\ t1=min (min t t') a==> h<t' /\ h< &1 /\ h< t /\ h<a`)
THEN RESA_TAC
THEN REMOVE_THEN "EM" (fun th-> MRESA1_TAC th `h:real`)
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[CROSS_SKEW]
THEN REWRITE_TAC[DOT_LNEG]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN REWRITE_TAC[GSYM DOT_LNEG]
THEN ONCE_REWRITE_TAC[GSYM CROSS_SKEW]
THEN RESA_TAC
THEN MRESA_TAC th3[`x:real^3`;`u:real^3`;`w:real^3`;]
THEN MRESA_TAC pos_in_aff_gt_2_1_fan [`x:real^3`;`u:real^3`;`w:real^3`;`h:real`]
THEN MRESAL_TAC aff_gt_2_1_cross_dotl_4point[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`;`(&1 - h) % u + h % w:real^3`][VECTOR_ARITH`((&1 - h) % u + h % w) - x=(&1 - h) % u + h % w - x`]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN STRIP_TAC
THEN MRESA_TAC aff_gt_inter_aff_gt [`(x:real^3)`;`(v:real^3)`;`(u:real^3)`]
THEN MP_TAC(SET_RULE`y IN aff_gt {x} {v, u} /\ aff_gt {x} {v, u} = aff_gt {x, v} {u} INTER aff_gt {x, u} {v}
==> y IN aff_gt {x, v} {u:real^3}`)
THEN RESA_TAC
THEN MRESAL_TAC aff_gt_2_1r_rcross_dotl_4point[`x:real^3`;`u:real^3`;`(&1 - h) % u + h % w:real^3`;`v:real^3`;`y:real^3`][VECTOR_ARITH`((&1 - h) % u + h % w) - x=(&1 - h) % u + h % w - x`]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN ONCE_REWRITE_TAC[CROSS_SKEW]
THEN ASM_REWRITE_TAC[DOT_LNEG]
THEN STRIP_TAC
THEN MRESAL_TAC condition_cross_dot_4point[`x:real^3`;`v:real^3`;`(&1 - h) % u + h % w:real^3`;`y:real^3`;`z:real^3` ][VECTOR_ARITH`((&1 - h) % u + h % w) - x=(&1 - h) % u + h % w - x`]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC condition_cross_dot_4point[`x:real^3`; `z:real^3`;`y:real^3` ;`v:real^3`;`(&1 - h) % u + h % w:real^3`][VECTOR_ARITH`((&1 - h) % u + h % w) - x=(&1 - h) % u + h % w - x`]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[CROSS_SKEW]
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[GSYM CROSS_RNEG]
THEN ONCE_REWRITE_TAC[GSYM CROSS_SKEW]
THEN REWRITE_TAC[DOT_LNEG;REAL_ARITH`--(--A)=A`]
THEN REMOVE_THEN "YEU" (fun th-> MRESA1_TAC th`h:real`)
THEN REMOVE_THEN "LINH" (fun th-> MRESA1_TAC th`h:real`)
THEN SET_TAC[]);;










let exists_dart_leads_into_edge_eq_topological1_component_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool y:real^3 z:real^3 v:real^3 u:real^3 w:real^3.
FAN(x,V,E) 
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ U IN topological_component_yfan (x,V,E)
/\ z IN U
/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ y IN aff_gt {x} {v,u} 
/\ y IN xfan(x,V,E)
/\ ~(y=x)
/\(!t. &0< t /\ t< &1==>   (&1-t)%y+t%z IN yfan(x,V,E))
/\ &0<((v-x) cross (y-x)) dot (z-x)

==> dart_leads_into x V E v u = U`,


REPEAT STRIP_TAC
THEN MRESA_TAC connected_component_of_faces_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`; `w:real^3`]
THEN FIND_ASSUM MP_TAC`fan80 (x:real^3,V,E)`
THEN REWRITE_TAC[fan80]
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`u:real^3`;`w:real^3`])
THEN MRESA_TAC fan_run_in_small_is_subset_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"HA")
THEN MRESA_TAC not_empty_rw_dart_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"EM")
THEN MRESA_TAC DART_LEADS_INTO[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;
`u:real^3`;`w:real^3`;]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"YEU")
THEN MRESA_TAC rw_dart_avoids_fan[`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"OI")
THEN MP_TAC(REAL_ARITH`&1> h'' /\ h'' > &0==> -- &1 < h'' /\ h''< &1 /\ -- &1 <= h'' /\ h''<= &1/\ &0 < h'' /\ h'' <= &1`) THEN RESA_TAC
THEN MRESA1_TAC ACS_BOUNDS_LT`h'':real`
THEN MRESAL_TAC ACS_MONO_LT[`&0`;`h'':real`][ACS_0;REAL_ARITH`-- &1 <= &0`]
THEN MRESA1_TAC COS_ACS `h'':real`
THEN ABBREV_TAC`h1= min h (min (h':real) (acs h''))/ &2`
THEN MP_TAC(REAL_ARITH`h1= min h (min (h':real) (acs h''))/ &2 /\ &0< h /\ &0<h' /\ &0< acs h'' /\ acs h''< pi/ &2==> &0< h1 /\ h1< h /\ h1<h' /\ h1<pi/ &2/\ h1< acs h'' /\ acs h'' <= pi /\ &0<= h1`)
THEN ASM_REWRITE_TAC[PI_WORKS] THEN STRIP_TAC
THEN MRESAL_TAC COS_MONO_LT[`h1:real`;`acs h'':real`][ACS_0;REAL_ARITH`-- &1 <= &0`]
THEN MP_TAC(REAL_ARITH`h''< cos h1==> h''<= cos h1`) THEN RESA_TAC
THEN REMOVE_THEN "EM"(fun th-> MRESAL1_TAC th `h1:real`[SET_RULE`~(A={})<=> ?y. y IN A`])
THEN REMOVE_THEN "YEU"(fun th-> MRESA_TAC th [`h1:real`;`y':real^3`])
THEN MRESA_TAC exists_rw_dart_inter_aff_gt1_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` v:real^3`;` u:real^3`;` w:real^3`;`h1:real`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"MA1")
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC point_in_yfan_and_point_in_xfan_indepent_fan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;` z:real^3`]
THEN MRESA_TAC exists_open_not_collinear[`(x:real^3)` ;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (v:real^3)`;`(u:real^3)`;`(w:real^3)`]
THEN SUBGOAL_THEN`(!h. &0 < h /\ h < t1 / &2 ==> ~collinear {x, v, (&1 - h) % u + h % w:real^3})`ASSUME_TAC
THENL[
POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"CHANGE")
THEN GEN_TAC THEN STRIP_TAC
THEN REMOVE_THEN "CHANGE"(fun th-> MRESA1_TAC th`h'''':real`)
THEN POP_ASSUM MATCH_MP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MP_TAC(REAL_ARITH`&0< t1 /\ t1<= &1==> &0< t1/ &2 /\ t1/ &2 < &1`)
THEN RESA_TAC
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`; `u:real^3`;`v:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN RESA_TAC
THEN MRESA_TAC condition_4point_aff_gt_1_2inter_aff_gt_1_2[`x:real^3`;`y:real^3`;`z:real^3`;`v:real^3`;` u:real^3`;`w:real^3`;`t1/ &2:real`]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"MA2")
THEN ABBREV_TAC`t2=min h ( min h''' t) / &2:real`
THEN MP_TAC(REAL_ARITH`t2=min h ( min h''' t) / &2:real /\ &0<h /\ &0<t /\ &0< h'''==> t2< h''' /\ t2< t /\ &0< t2/\ t2< h`)
THEN RESA_TAC
THEN REMOVE_THEN "MA1"(fun th-> MRESA1_TAC th `t2:real`)
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN REWRITE_TAC[SET_RULE`~(A={})<=> ?y1. y1 IN A`;INTER;IN_ELIM_THM]
THEN RESA_TAC
THEN REMOVE_THEN "MA2"(fun th-> MRESA1_TAC th `t2:real`)
THEN MRESA_TAC nonsetedge_fully_surround_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`]
THEN MRESA_TAC point_in_yfan_not_x_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;` z:real^3`]
THEN MP_TAC(SET_RULE`~(x=z) /\ ~(y=x)==> DISJOINT {x} {y,z:real^3}`)
THEN RESA_TAC
THEN MRESA_TAC aff_gt_subset_component_y_fan [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;`z:real^3`]
THEN MP_TAC(SET_RULE`~(aff_gt {x} {y, z} INTER aff_gt {x} {v, (&1 - t2) % u + t2 % w} = {})/\
aff_gt {x} {y, z} SUBSET U==> ~(U INTER aff_gt {x:real^3} {v, (&1 - t2) % u + t2 % w} = {})`)
THEN ASM_REWRITE_TAC[]
THEN REWRITE_TAC[SET_RULE`~(A={})<=> ?y2. y2 IN A`;INTER;IN_ELIM_THM]
THEN RESA_TAC
THEN MRESA_TAC expand_element_in_topological_component_yfan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`U:real^3->bool`;`y2:real^3`]
THEN MRESA_TAC dart_leads_into_fan_in_topological_component_yfan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`]
THEN MP_TAC(SET_RULE`y1 IN rw_dart_fan x V E (x,u,w,v) (cos h1)
/\ rw_dart_fan x V E (x,u,w,v) (cos h1) SUBSET (dart_leads_into x V E u w)
==> y1 IN (dart_leads_into x V E u (w:real^3))`)
THEN RESA_TAC
THEN MRESA_TAC expand_element_in_topological_component_yfan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`(dart_leads_into x V E u w):real^3->bool`;`y1:real^3`]
THEN MRESA_TAC zpoint_in_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;`y2:real^3`]
THEN MRESA_TAC zpoint_in_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(dart_leads_into x V E u w):real^3->bool`;`y1:real^3`]
THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_EQ_EQ]
THEN MRESA_TAC CONVEX_AFF_GT[`{x:real^3}`;`{v, (&1 - t2) % u + t2  % w:real^3}`]
THEN MRESA_TAC CONVEX_CONNECTED[`aff_gt {x} {v, (&1 - t2) % u + t2 % w}:real^3->bool`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[CONNECTED_IFF_CONNECTED_COMPONENT]
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th [`y1:real^3`;`y2:real^3`])
THEN REMOVE_THEN "HA" (fun th-> MRESA1_TAC th `t2:real`)
THEN MRESA_TAC CONNECTED_COMPONENT_OF_SUBSET[`aff_gt {x} {v, (&1 - t2) % u + t2 % w:real^3}:real^3-> bool`;`yfan (x:real^3,(V:real^3->bool),(E:(real^3->bool)->bool)):real^3->bool`;`y1:real^3`;`y2:real^3`]
]);;




let JUTSTKG=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool.
FAN(x,V,E) 
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ U IN topological_component_yfan (x,V,E)
==> ?v u. {v,u} IN E /\ dart_leads_into x V E v u = U`,

REPEAT STRIP_TAC
THEN MRESA_TAC exists_point_in_component_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)` ;`U:real^3->bool`]
THEN MRESA_TAC connect_insidepoint_to_bound_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)` ;`U:real^3->bool`;`z:real^3`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th-> MP_TAC th THEN ASSUME_TAC th)
THEN REWRITE_TAC[xfan;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC expand_edge_graph_fan [`(x:real^3)`;` (V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(e:real^3->bool)`]
THEN POP_ASSUM MP_TAC
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[IN]
THEN RESA_TAC
THEN SUBGOAL_THEN `{v,w:real^3} IN E`ASSUME_TAC
THENL(*1*)[POP_ASSUM (fun th-> ASM_REWRITE_TAC[SYM(th);IN]);(*1*)
MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (v:real^3)`]
THEN MRESAL_TAC aff_ge_eq_aff_gt_union_aff_ge[`x:real^3`;`v:real^3`;`w:real^3`][UNION;IN_ELIM_THM]
THEN STRIP_TAC
THENL(*2*)[
STRIP_TAC
THEN MRESA_TAC point_in_yfan_and_point_in_xfan_indepent_fan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;` z:real^3`]
THEN DISJ_CASES_TAC(REAL_ARITH`&0 < ((v - x) cross (y - x)) dot (z - x) \/ &0< --(((v - x) cross (y - x)) dot (z - x)) \/  ((v - x:real^3) cross (y - x)) dot (z - x)= &0`)
THENL(*3*)[
MRESA_TAC INVERSE1_SIGMA_FAN[`x:real^3`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`w:real^3`]
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA1_TAC th `v:real^3`)
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `v:real^3`)
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN STRIP_TAC
THEN MRESA_TAC exists_dart_leads_into_edge_eq_topological1_component_fan[`x:real^3`;`(V:real^3->bool)` ;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;`z:real^3`;`v:real^3`;`w:real^3`;`inverse1_sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (w:real^3) (v:real^3)`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN RESA_TAC
THEN EXISTS_TAC`v:real^3`
THEN EXISTS_TAC`w:real^3`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[];(*3*)

POP_ASSUM MP_TAC
THEN STRIP_TAC
THENL(*4*)[
MRESA_TAC aff_gt_1_2_cross_dotr_4point_neg[`x:real^3`;`y:real^3`;`z:real^3`;`v:real^3`;`w:real^3`;]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN RESA_TAC
THEN MRESA_TAC INVERSE1_SIGMA_FAN[`x:real^3`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`v:real^3`]
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA1_TAC th `w:real^3`)
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th-> MRESA1_TAC th `w:real^3`)
THEN STRIP_TAC
THEN MRESA_TAC exists_dart_leads_into_edge_eq_topological1_component_fan[`x:real^3`;`(V:real^3->bool)` ;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;`z:real^3`;`w:real^3`;`v:real^3`;`inverse1_sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (w:real^3)`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN RESA_TAC
THEN EXISTS_TAC`w:real^3`
THEN EXISTS_TAC`v:real^3`
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[];(*4*)

MRESA_TAC aff_gt_1_2_cross_dotr_4point_zero[`x:real^3`;`y:real^3`;`z:real^3`;`v:real^3`;`w:real^3`;]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN RESA_TAC
THEN MRESA_TAC not_azim_points1_in_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;`z:real^3`;`v:real^3`;`w:real^3`]
THEN MRESA_TAC not_azim_points1_in_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;`z:real^3`;`w:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MP_TAC(REAL_ARITH`~(azim x y z v = &0) /\ &0<= azim x y z v==> &0< azim x y z (v:real^3) `)
THEN ASM_REWRITE_TAC[azim]
THEN STRIP_TAC
THEN MRESA_TAC properties_of_collinear4_points_fan[`x:real^3`;`v:real^3`;`w:real^3`;`y:real^3`]
THEN MRESA_TAC properties_of_collinear4_points_fan[`x:real^3`;`w:real^3`;`v:real^3`;`y:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN STRIP_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`azim x y z (v:real^3)< pi \/ pi<= azim x y z (v:real^3)`)
THENL(*5*)[
MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`y:real^3`; `v:real^3`;`z:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;(*5*)

POP_ASSUM MP_TAC
THEN MRESA_TAC aff_gt2_subset_aff_ge [`x:real^3`;`w:real^3`; `v:real^3`;`y:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th)] THEN ASSUME_TAC th)
THEN STRIP_TAC
THEN MRESA_TAC sum5_azim_fan[`x:real^3`;`y:real^3`;`z:real^3`;`w:real^3`;`v:real^3`]
THEN MP_TAC(REAL_ARITH`azim x y z v = azim x y z w + pi/\
azim x y z v < &2 * pi/\ ~(azim x y z w = &0) /\ &0 <= azim x y z w 
==>  &0< azim x y z w /\ azim x y z w < pi
`)
THEN ASM_REWRITE_TAC[azim]
THEN STRIP_TAC
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`y:real^3`; `w:real^3`;`z:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC](*5*)](*4*)](*3*);(*2*)


STRIP_TAC
THEN MRESA_TAC exists_dart_leads_into_edge_eq_topological_component_fan[`x:real^3`;`(V:real^3->bool)` ;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;`z:real^3`;`v:real^3`]
THEN EXISTS_TAC `v:real^3`
THEN EXISTS_TAC `w':real^3`
THEN ASM_REWRITE_TAC[];
STRIP_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (v:real^3)`;
` (w:real^3)`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MRESA_TAC exists_dart_leads_into_edge_eq_topological_component_fan[`x:real^3`;`(V:real^3->bool)` ;`(E:(real^3->bool)->bool)`;`U:real^3->bool`;`y:real^3`;`z:real^3`;`w:real^3`]
THEN EXISTS_TAC `w:real^3`
THEN EXISTS_TAC `w':real^3`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]]]);;


(**************KVQWYDL*******************)



let AFF_GT_3_1 = prove
 (`!x v u w.
        DISJOINT {x,v,u} {w}
        ==> aff_gt {x,v,u} {w} =
             {y | ?t1 t2 t3 t4.
                     &0 < t4 /\
                     t1 + t2 +t3 +t4 = &1 /\
                     y = t1 % x + t2 % v + t3 % u +t4 % w }`,
  AFF_TAC);;


let AFF_GT_1_3 = prove
 (`!x v u w.
        DISJOINT {x} {v,u,w}
        ==> aff_gt {x} {v,u,w} =
             {y | ?t1 t2 t3 t4.
                     &0 < t2 /\ &0< t3 /\ &0<t4 /\
                     t1 + t2 +t3 +t4 = &1 /\
                     y = t1 % x + t2 % v + t3 % u +t4 % w }`,
  AFF_TAC);;
let AFF_GE_1_3 = prove
 (`!x v u w.
        DISJOINT {x} {v,u,w}
        ==> aff_ge {x} {v,u,w} =
             {y | ?t1 t2 t3 t4.
                     &0 <= t2 /\ &0<= t3 /\ &0<= t4 /\
                     t1 + t2 +t3 +t4 = &1 /\
                     y = t1 % x + t2 % v + t3 % u +t4 % w }`,
  AFF_TAC);;


let notcoplanar_disjoint=prove(`!x  v u w:real^3.
~coplanar {x,v,u,w} 
==> ~(x=v) /\ ~(x=u) /\ ~(x=w)/\ ~(v=u) /\ ~(v=w) /\ ~(u=w)`,
REPEAT GEN_TAC
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> MP_TAC th THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={B,C,D,A}`] THEN DISCH_TAC THEN MP_TAC th THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={C,D,A,B}`] THEN DISCH_TAC THEN ASSUME_TAC th)
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`v:real^3`;`u:real^3`;`w:real^3`;`x:real^3`]
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`u:real^3`;`w:real^3`;`x:real^3`;`v:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`w:real^3`]
THEN MRESA_TAC th3[`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC th3[`u:real^3`;`w:real^3`;`x:real^3`]);;


let notcoplanar_disjoints=prove(`!x  v u w:real^3.
~coplanar {x,v,u,w} 
==> DISJOINT{x,v,u} {w} /\ DISJOINT{x,u,w} {v} /\ DISJOINT{x,w,v} {u} /\ DISJOINT{x} {v,u,w}/\ DISJOINT {x,u} {v,w} /\ DISJOINT {x} {v,u} /\ DISJOINT {x} {u,w} /\ DISJOINT {x} {w,v}  `,
REPEAT STRIP_TAC
THEN MRESA_TAC notcoplanar_disjoint[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN ASM_TAC THEN SET_TAC[]);;


let coplanar_imp_continuous_collinear=prove(`!x:real^3  v:real^3 u:real^3 w:real^3.
 ~(coplanar{x,v,u,w})
==>(!t:real.  ~(t= &0) ==> ~collinear {x,v,(&1-t)%u+t%w} )`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC continuous_coplanar_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th`t:real`)
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`(&1 - t) % u + t % w:real^3`]);;




let aff_gt_1_3_eq_unions_aff_gt_1_2=prove(`!x  v u w:real^3.
~coplanar {x,v,u,w}
==>
aff_gt {x} {v,u,w} = UNIONS {aff_gt{x} {v,(&1-a)%u+ a % w}| &0<a /\ a< &1}`,
REPEAT STRIP_TAC
THEN MRESA_TAC notcoplanar_disjoints[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC notcoplanar_disjoint[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC AFF_GT_1_3[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN REWRITE_TAC[EXTENSION;IN_ELIM_THM;UNION]
THEN REDUCE_VECTOR_TAC
THEN GEN_TAC
THEN EQ_TAC
THENL[
STRIP_TAC
THEN ASM_REWRITE_TAC[UNIONS;IN_ELIM_THM]
THEN MP_TAC(REAL_ARITH`&0< t3 /\ &0< t4==> &0< t3+t4/\ ~(t3+t4= &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV `t3+t4:real`
THEN MRESA_TAC REAL_LT_MUL [`inv(t3+t4:real)`;`t4:real`]
THEN MRESA_TAC REAL_LT_MUL [`inv(t3+t4:real)`;`t3:real`]
THEN MRESA1_TAC REAL_MUL_LINV `t3+t4:real`
THEN POP_ASSUM (fun th-> MP_TAC th THEN REWRITE_TAC[REAL_ARITH`A*(B+C)= &1<=> A*B = &1 - A*C`]
THEN STRIP_TAC
THEN MP_TAC(REAL_ARITH`inv (t3 + t4) * t3  = &1 -inv (t3 + t4) * t4/\ &0< inv (t3 + t4) * t3 ==>
inv (t3 + t4) *  t4 < &1 `)
THEN RESA_TAC THEN ASSUME_TAC th)
THEN MP_TAC(REAL_ARITH`&0<inv (t3 + t4) *  t4==> ~(inv (t3 + t4) *  t4= &0)`) THEN RESA_TAC
THEN EXISTS_TAC`aff_gt {x:real^3} {v, (&1 - inv (t3 + t4:real) * t4) %u + (inv (t3 + t4) * t4) % w}`
THEN STRIP_TAC
THENL[
EXISTS_TAC`inv(t3 + t4:real) * t4`
THEN ASM_REWRITE_TAC[];

 MRESA_TAC coplanar_imp_continuous_collinear[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th`inv (t3 + t4:real) *  t4`)
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`(&1 - inv (t3 + t4) * t4) % u + (inv (t3 + t4) * t4) % w:real^3`]
THEN MRESAL_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`(&1 - inv (t3 + t4) * t4) % u + (inv (t3 + t4) * t4) % w:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3+ t4:real`
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(t3 + t4) % ((&1 - inv (t3 + t4) * t4) % u + (inv (t3 + t4) * t4) % w)
=(t3 + t4 - (inv (t3 + t4) *(t3+t4))* t4) % u + ((inv (t3 + t4)*(t3+t4)) * t4) % w`]
THEN VECTOR_ARITH_TAC];


REWRITE_TAC[UNIONS;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(REAL_ARITH`&0<a==> ~(a= &0)`) THEN RESA_TAC
THEN MRESA_TAC coplanar_imp_continuous_collinear[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th`a:real`)
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`(&1 - a) % u + (a) % w:real^3`]
THEN MRESAL_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`(&1 - a) % u + a % w:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3 *(&1-a):real`
THEN EXISTS_TAC`t3 * a:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`t1 + t2 + t3 * (&1 - a) + t3 * a=t1+t2+t3`;
VECTOR_ARITH`t1 % x + t2 % v + t3 % ((&1 - a) % u + a % w) =
 t1 % x + t2 % v + (t3 * (&1 - a)) % u + (t3 * a) % w:real^3`]
THEN STRIP_TAC
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC]);;













let aff_gt_1_3_subset_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
==>  (aff_gt{x} {v,u,w} SUBSET yfan (x,V,E))`,
REPEAT STRIP_TAC
THEN POP_ASSUM (fun th -> MP_TAC th
THEN REWRITE_TAC[fan80] THEN ASSUME_TAC th)
THEN DISCH_TAC
THEN POP_ASSUM (fun th -> MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC "YEU EM"))
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESAL_TAC aff_gt_1_3_eq_unions_aff_gt_1_2[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`][UNIONS;IN_ELIM_THM;yfan;xfan]
THEN MRESA_TAC not_cut_in_edges_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;]
THEN POP_ASSUM MP_TAC
THEN SET_TAC[]);;



let aff_gt_1_3_subset_dart_leads_into_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E 
/\ sigma_fan x V E u w = v
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
==>  aff_gt{x} {v,u,w} SUBSET dart_leads_into x V E u w`,
REPEAT STRIP_TAC
THEN POP_ASSUM (fun th -> MP_TAC th
THEN REWRITE_TAC[fan80] THEN ASSUME_TAC th)
THEN DISCH_TAC
THEN POP_ASSUM (fun th -> MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC "YEU EM"))
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESAL_TAC aff_gt_1_3_eq_unions_aff_gt_1_2[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`][UNIONS;IN_ELIM_THM;yfan;xfan]
THEN MRESA_TAC not_empty_rw_dart_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"EM")
THEN MRESA_TAC DART_LEADS_INTO[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;
`u:real^3`;`w:real^3`;]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"YEU")
THEN MRESA_TAC rw_dart_avoids_fan[`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"OI")
THEN MP_TAC(REAL_ARITH`&1> h' /\ h' > &0==> -- &1 < h' /\ h'< &1 /\ -- &1 <= h' /\ h'<= &1/\ &0 < h' /\ h' <= &1`) THEN RESA_TAC
THEN MRESA1_TAC ACS_BOUNDS_LT`h':real`
THEN MRESAL_TAC ACS_MONO_LT[`&0`;`h':real`][ACS_0;REAL_ARITH`-- &1 <= &0`]
THEN MRESA1_TAC COS_ACS `h':real`
THEN ABBREV_TAC`h1= min (h:real) (acs h')/ &2`
THEN MP_TAC(REAL_ARITH`h1= min (h:real) (acs h')/ &2 /\ &0<h /\ &0< acs h' /\ acs h'< pi/ &2==> &0< h1 /\ h1< h /\ h1<pi/ &2/\ h1< acs h' /\ acs h' <= pi /\ &0<= h1`)
THEN ASM_REWRITE_TAC[PI_WORKS] THEN STRIP_TAC
THEN MRESAL_TAC COS_MONO_LT[`h1:real`;`acs h':real`][ACS_0;REAL_ARITH`-- &1 <= &0`]
THEN MP_TAC(REAL_ARITH`h'< cos h1==> h'<= cos h1`) THEN RESA_TAC
THEN REMOVE_THEN "EM"(fun th-> MRESAL1_TAC th `h1:real`[SET_RULE`~(A={})<=> ?y. y IN A`])
THEN REMOVE_THEN "YEU"(fun th-> MRESA_TAC th [`h1:real`;`y:real^3`])
THEN POP_ASSUM(fun th -> REWRITE_TAC[SYM(th);IN_ELIM_THM;topological_component_yfan;] THEN ASSUME_TAC th) 
THEN MRESA_TAC exists_rw_dart_inter_aff_gt1_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` v:real^3`;` u:real^3`;` w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th`h1:real`)
THEN POP_ASSUM MP_TAC
THEN ABBREV_TAC`a1=min h'' (&1) / &2`
THEN MP_TAC(REAL_ARITH`&0< h'' /\ a1=min h'' (&1) / &2 ==> &0< a1 /\ a1< h'' /\ a1< &1`)
THEN RESA_TAC
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> MRESAL1_TAC th`a1:real`[SET_RULE`~(A={})<=> ?y. y IN A`;INTER;IN_ELIM_THM])
THEN MP_TAC(SET_RULE`y' IN rw_dart_fan x V E (x,u,w,v) (cos h1)/\ rw_dart_fan x V E (x,u,w,v) (cos h1) SUBSET dart_leads_into x V E u w
==> y' IN dart_leads_into x V E u w`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`y' IN aff_gt {x} {v, (&1 - a1) % u + a1 % w:real^3} /\ &0< a1 /\  (a1:real) < &1
==> y' IN {x' | ?u'. (?a. (&0 < a /\ a < &1) /\
                u' = aff_gt {x} {v, (&1 - a) % u + a % w}) /\
           x' IN u'}`)
THEN RESA_TAC
THEN MRESA_TAC dart_leads_into_fan_in_topological_component_yfan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC expand_element_in_topological_component_yfan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`(dart_leads_into x V E u w):real^3->bool`;`y':real^3`]
THEN MRESA_TAC CONVEX_AFF_GT[`{x:real^3}`;`{v, u, w:real^3}`]
THEN MRESA_TAC CONVEX_CONNECTED[`aff_gt {x} {v, u,w}:real^3->bool`]
THEN MRESA_TAC aff_gt_1_3_subset_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` v:real^3`;` u:real^3`;` w:real^3`]
THEN MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL
THEN ASM_REWRITE_TAC[]);;






let inter_aff_gt_3_1_is_aff_gt_1_3=prove(`!x v u w:real^3.
~coplanar {x,v,u,w}
==>
aff_gt {x,v,u} {w} INTER aff_gt {x,u,w} {v} INTER aff_gt {x,w,v} {u}=aff_gt {x} {v,u,w}`,
GEOM_ORIGIN_TAC `x:real^3`
THEN REPEAT STRIP_TAC
THEN MRESA_TAC notcoplanar_disjoints[`(vec 0):real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC notcoplanar_disjoint[`(vec 0):real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC AFF_GT_3_1[`(vec 0):real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC AFF_GT_3_1[`(vec 0):real^3`;`u:real^3`;`w:real^3`;`v:real^3`]
THEN MRESA_TAC AFF_GT_3_1[`(vec 0):real^3`;`w:real^3`;`v:real^3`;`u:real^3`]
THEN MRESA_TAC AFF_GT_1_3[`(vec 0):real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC NOT_COPLANAR_0_4_IMP_INDEPENDENT[`v:real^3`;`u:real^3`;`w:real^3`]
THEN REWRITE_TAC[EXTENSION;IN_ELIM_THM;INTER]
THEN REDUCE_VECTOR_TAC
THEN GEN_TAC
THEN EQ_TAC
THENL[
STRIP_TAC
THEN POP_ASSUM MP_TAC 
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t2 % v + t3 % u + t4 % w = t2'' % w + t3'' % v + t4'' % u <=> (t2-t3'') % v + (t3-t4'') % u + (t4-t2'') % w= vec 0`]
THEN STRIP_TAC
THEN MRESAL_TAC INDEPENDENT_3[`v:real^3`;`u:real^3`;`w:real^3`;`t2-t3'':real`; `t3-t4'':real`; `t4 -t2'':real`][REAL_ARITH`A-B= &0<=> A=B`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC 
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t2 % v + t3 % u + t4 % w = t2' % u + t3' % w + t4' % v <=> (t2-t4') % v + (t3-t2') % u + (t4-t3') % w= vec 0`]
THEN STRIP_TAC
THEN MRESAL_TAC INDEPENDENT_3[`v:real^3`;`u:real^3`;`w:real^3`;`t2-t4':real`; `t3-t2':real`; `t4 -t3':real`][REAL_ARITH`A-B= &0<=> A=B`]
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`t1'':real`
THEN EXISTS_TAC`t3'':real`
THEN EXISTS_TAC`t4'':real`
THEN EXISTS_TAC`t2'':real`
THEN ASM_REWRITE_TAC[REAL_ARITH`t1+t2+t3+t4=t1+t4+t2+t3:real`]
THEN ASM_TAC
THEN REAL_ARITH_TAC;

STRIP_TAC
THEN STRIP_TAC
THENL[EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3:real`
THEN EXISTS_TAC`t4:real`
THEN ASM_REWRITE_TAC[];

STRIP_TAC
THENL[
EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t3:real`
THEN EXISTS_TAC`t4:real`
THEN EXISTS_TAC`t2:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`t1  + t3 + t4+ t2:real=t1 + t2 + t3 + t4`]
THEN VECTOR_ARITH_TAC;
EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t4:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`t1  + t4+ t2+ t3:real=t1 + t2 + t3 + t4`]
THEN VECTOR_ARITH_TAC]]]);;

let coplanar_cross_dot=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
~coplanar {x,v,u,v1}
==> ~(((v-x) cross (u-x)) dot (v1-x)= &0)`,
REPEAT GEN_TAC
THEN MATCH_MP_TAC MONO_NOT
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN ONCE_REWRITE_TAC[DOT_SYM;]
THEN REWRITE_TAC[DOT_CROSS_DET]
THEN ONCE_REWRITE_TAC[ COPLANAR_DET_EQ_0]
THEN ASM_REWRITE_TAC[]);;

let CRAMER_LEMMA1 = prove
 (`!A:real^N^N x:real^N k:num.
        1 <= k /\ k <= dimindex(:N)
        ==> det((lambda i j. if j = k then (A**x)$i else A$i$j):real^N^N) =
            x$k * det(A)`,
  REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[MATRIX_MUL_VSUM] THEN
  FIRST_ASSUM(MP_TAC o SYM o SPECL [`transp(A:real^N^N)`; `x:real^N`] o
              MATCH_MP CRAMER_LEMMA_TRANSP) THEN
  REWRITE_TAC[DET_TRANSP] THEN DISCH_THEN SUBST1_TAC THEN
  GEN_REWRITE_TAC LAND_CONV [GSYM DET_TRANSP] THEN AP_TERM_TAC THEN
  ASM_SIMP_TAC[CART_EQ; transp; LAMBDA_BETA; MATRIX_MUL_VSUM; row; column;
        COND_COMPONENT; VECTOR_MUL_COMPONENT; VSUM_COMPONENT]);;
let aff_gt_3_1_rep_cross_dot=prove(`!x:real^3  v:real^3 u:real^3 w:real^3.
~coplanar {x,v,u,w} 
/\ &0< ((v-x) cross (u-x)) dot (w-x)

==> aff_gt {x,v,u} {w} ={y:real^3|  &0< (((v-x) cross (u-x)) dot (y-x)) }`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC notcoplanar_disjoints[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN DISCH_THEN(LABEL_TAC"YEU")
THEN MRESA_TAC AFF_GT_3_1[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN REWRITE_TAC[EXTENSION;IN_ELIM_THM]
THEN GEN_TAC
THEN EQ_TAC
THENL[STRIP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`((t1 % x + t2 % v + t3 % u + t4 % w) - x)=((t1+t2+t3+t4) - &1) % x + t2 % (v-x) + t3 % (u-x) + t4 % (w - x)`; REAL_ARITH`&1- &1= &0`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[CROSS_LNEG;CROSS_LMUL;CROSS_LADD;CROSS_REFL;DOT_RMUL;DOT_RADD;DOT_CROSS_SELF;]
THEN REDUCE_ARITH_TAC
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[];
DISCH_THEN(LABEL_TAC"ME")
THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`]properties_coordinate) THEN RESA_TAC
THEN MRESA_TAC ORTHONORMAL_IMP_SPANNING[`e1_fan (x:real^3) (v:real^3) (u:real^3)`;`e2_fan (x:real^3)( v:real^3) (u:real^3)`;`e3_fan (x:real^3) (v:real^3) (u:real^3)`;]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[SPAN_3;EXTENSION] 
 THEN DISCH_TAC THEN POP_ASSUM(fun th-> MP_TAC(ISPEC`(x':real^3)-(x:real^3)`th) THEN ASSUME_TAC(th))
THEN POP_ASSUM(fun th-> MP_TAC(ISPEC`(w:real^3)-(x:real^3)`th)THEN ASSUME_TAC(th))
THEN POP_ASSUM(fun th-> MP_TAC(ISPEC`(u:real^3)-(x:real^3)`th)THEN ASSUME_TAC(th))
THEN POP_ASSUM(fun th-> MP_TAC(ISPEC`(v:real^3)-(x:real^3)`th))
THEN REWRITE_TAC[SET_RULE`(a:real^3) IN (:real^3)`;IN_ELIM_THM;VECTOR_ARITH`A-B=C<=>A=C+B:real^3`] 
THEN REPEAT STRIP_TAC
THEN ABBREV_TAC`e1=e1_fan x v u:real^3`
THEN ABBREV_TAC`e2=e2_fan x v u:real^3`
THEN ABBREV_TAC`e3=e3_fan x v u:real^3`
THEN REMOVE_THEN"YEU" MP_TAC
THEN MRESA_TAC ORTHONORMAL_CROSS[`e1:real^3`;`e2:real^3`;`e3:real^3`;]
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1 % x +
     t2 % ((u' % e1 + v' % e2 + w' % e3) + x) +
     t3 % ((u'' % e1 + v'' % e2 + w'' % e3) + x) +
     t4 % ((u''' % e1 + v''' % e2 + w''' % e3) + x)
=     (t2 * u' + t3 * u'' +t4 * u''') % e1 + (t2 * v'+t3*v''+ t4 * v''') % e2 + (t2 * w'+ t3 * w''+ t4 * w''') % e3 +(t1+t2+t3+t4) % x :real^3`;VECTOR_ARITH`((u' % e1 + v' % e2 + w' % e3) + x) - x=u' % e1 + v' % e2 + w' % e3`;CROSS_LMUL;CROSS_RMUL;CROSS_LADD;CROSS_RADD;CROSS_REFL;]
THEN ONCE_REWRITE_TAC[CROSS_SKEW]
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN FIND_ASSUM MP_TAC`orthonormal e1 e2 (e3:real^3)`
THEN REWRITE_TAC[orthonormal]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[DOT_LADD;DOT_RADD;DOT_LMUL;DOT_RMUL;DOT_LNEG]
THEN ONCE_REWRITE_TAC[DOT_SYM]
THEN ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN ABBREV_TAC`a1=vector[u':real;u'':real;u''':real]:real^3`
THEN ABBREV_TAC`a2=vector[v':real;v'':real;v''':real]:real^3`
THEN ABBREV_TAC`a3=vector[w':real;w'':real;w''':real]:real^3`
THEN ABBREV_TAC`A=vector[a1;a2;a3:real^3]:real^3^3`
THEN ABBREV_TAC`b=vector[u'''':real;v'''':real;w'''':real]:real^3`
THEN STRIP_TAC
THEN SUBGOAL_THEN`&0< det(A:real^3^3)`ASSUME_TAC
THENL[	
EXPAND_TAC"A"
THEN EXPAND_TAC "a1"
THEN EXPAND_TAC "a2"
THEN EXPAND_TAC "a3"
THEN REWRITE_TAC[DET_3;VECTOR_3]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MP_TAC(REAL_ARITH`&0< det (A:real^3^3)==> ~(det A= &0)`) THEN RESA_TAC
THEN MRESA_TAC CRAMER[`A:real^3^3`;]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[SWAP_FORALL_THM]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`b:real^3`])
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[MESON[]
   ` (!x. p(x) <=> x = a)  <=> (?x. p(x)) /\ (!x. p(x) ==> x = a)`]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "LINH1")
THEN DISCH_THEN(LABEL_TAC "LINH2")
THEN REMOVE_THEN "LINH1" MP_TAC
THEN MRESAL_TAC MATRIX_VECTOR_MUL_COMPONENT[`A:real^3^3`;`x'':real^3`][DIMINDEX_3]
THEN POP_ASSUM(fun th-> MRESAL1_TAC th`1:num`[ARITH_RULE`1<=1/\ 1<=3`;LAMBDA_BETA;VECTOR_3] THEN ASSUME_TAC th)
THEN POP_ASSUM(fun th-> MRESAL1_TAC th`2:num`[ARITH_RULE`1<=2/\ 2<=3`;VECTOR_3] THEN ASSUME_TAC th)
THEN POP_ASSUM(fun th-> MRESAL1_TAC th`3:num`[ARITH_RULE`1<=3/\ 3<=3`;VECTOR_3;])
THEN DISCH_TAC
THEN POP_ASSUM(fun th -> MP_TAC th THEN ASSUME_TAC th)
THEN GEN_REWRITE_TAC(LAND_CONV o DEPTH_CONV)[VECTOR_3;CART_EQ;]
THEN REWRITE_TAC[DIMINDEX_3]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESAL1_TAC th`1:num`[ARITH_RULE`1<=1/\ 1<=3`;DOT_SYM;] THEN ASSUME_TAC th)
THEN POP_ASSUM(fun th-> MRESAL1_TAC th`2:num`[ARITH_RULE`1<=2/\ 2<=3`;DOT_SYM;] THEN ASSUME_TAC th)
THEN POP_ASSUM(fun th-> MRESAL1_TAC th`3:num`[ARITH_RULE`1<=3/\ 3<=3`;DOT_SYM;])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC"A"
THEN EXPAND_TAC "a1"
THEN EXPAND_TAC "a2"
THEN EXPAND_TAC "a3"
THEN EXPAND_TAC "b"
THEN REWRITE_TAC[VECTOR_3;DOT_3]
THEN DISCH_THEN(LABEL_TAC"MA")
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`&1- (x'':real^3)$1 - x''$2 -x''$3`
THEN EXISTS_TAC`(x'':real^3)$1`
THEN EXISTS_TAC`(x'':real^3)$2`
THEN EXISTS_TAC`(x'':real^3)$3`
THEN ASM_REWRITE_TAC[REAL_ARITH` &1 - x''$1 - x''$2 - x''$3 + x''$1 + x''$2 + x''$3= &1`;VECTOR_ARITH`(u'''' % e1 + v'''' % e2 + w'''' % e3) + x =
 u'''' % e1 + v'''' % e2 + w'''' % e3 + &1 % x`]
THEN REMOVE_THEN "MA" MP_TAC
THEN RESA_TAC
THEN MRESAL_TAC CRAMER_LEMMA1[`A:real^3^3`;`x'':real^3`;`3`][ARITH_RULE`1<=3/\ 3<=3`;DIMINDEX_3;]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN DISCH_THEN(LABEL_TAC"LINH3")
THEN ABBREV_TAC`b1=vector[u':real;u'':real;u'''':real]:real^3`
THEN ABBREV_TAC`b2=vector[v':real;v'':real;v'''':real]:real^3`
THEN ABBREV_TAC`b3=vector[w':real;w'':real;w'''':real]:real^3`
THEN ABBREV_TAC`B=vector[b1;b2;b3:real^3]:real^3^3`
THEN SUBGOAL_THEN`(lambda i j. if j = 3 then (b:real^3)$i else (A:real^3^3)$i$j):real^3^3=B` ASSUME_TAC
THENL[ONCE_ASM_SIMP_TAC[CART_EQ;]
THEN ONCE_ASM_SIMP_TAC[CART_EQ;]
THEN ASM_SIMP_TAC[LAMBDA_BETA;DIMINDEX_3]
THEN REPEAT STRIP_TAC
THEN COND_CASES_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`1<=i /\ i<= 3==> i=1 \/ i=2 \/ i=3`)
THEN RESA_TAC
THEN EXPAND_TAC"B"
THEN EXPAND_TAC "b1"
THEN EXPAND_TAC "b2"
THEN EXPAND_TAC "b3"
THEN EXPAND_TAC "b"
THEN MP_TAC(ARITH_RULE`1<=i' /\ i'<= 3/\ ~(i'=3)==> i'=1 \/ i'=2`)
THEN EXPAND_TAC "A"
THEN REWRITE_TAC[VECTOR_3]
THEN RESA_TAC
THEN EXPAND_TAC "b1"
THEN EXPAND_TAC "b2"
THEN EXPAND_TAC "b3"
THEN EXPAND_TAC "a3"
THEN EXPAND_TAC "a1"
THEN EXPAND_TAC "a2"
THEN REWRITE_TAC[VECTOR_3];
REMOVE_THEN"LINH3" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN REMOVE_ASSUM_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN`&0< det(B:real^3^3)`ASSUME_TAC
THENL[REMOVE_ASSUM_TAC
THEN EXPAND_TAC"B"
THEN EXPAND_TAC "b1"
THEN EXPAND_TAC "b2"
THEN EXPAND_TAC "b3"
THEN REWRITE_TAC[DET_3;VECTOR_3]
THEN REMOVE_THEN "ME" MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`((u' % e1 + v' % e2 + w' % e3) + x) - x=u' % e1 + v' % e2 + w' % e3`;CROSS_LMUL;CROSS_RMUL;CROSS_LADD;CROSS_RADD;CROSS_REFL;]
THEN ONCE_REWRITE_TAC[CROSS_SKEW]
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[DOT_LADD;DOT_RADD;DOT_LMUL;DOT_RMUL;DOT_LNEG]
THEN ONCE_REWRITE_TAC[DOT_SYM]
THEN ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN REAL_ARITH_TAC;
POP_ASSUM MP_TAC
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`det (A:real^3^3)`
THEN MRESA1_TAC REAL_MUL_LINV`det (A:real^3^3)`
THEN MRESAL_TAC REAL_LT_MUL[`inv(det(A:real^3^3))`;`(x'':real^3)$3 * det(A:real^3^3)`][REAL_ARITH`A*(B*C)=(A*C)*B`;REAL_ARITH`&1*A=A`]]]]]);;




let OPEN_AFF_GT_1_3=prove(`!x v u w:real^3.
~coplanar {x,v,u,w}
==>
open (aff_gt {x} {v,u,w})`,

 GEOM_ORIGIN_TAC `x:real^3` THEN
 REPEAT STRIP_TAC
THEN POP_ASSUM (fun th-> MP_TAC th THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,D,B}`] THEN DISCH_TAC  THEN
MP_TAC th THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,B,C}`] THEN DISCH_TAC THEN ASSUME_TAC th)
THEN MRESA_TAC inter_aff_gt_3_1_is_aff_gt_1_3[`vec 0:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN MRESAL_TAC coplanar_cross_dot[`vec 0:real^3`;`v:real^3`;`u:real^3`;`w:real^3`][VECTOR_ARITH`A- vec  0=A`]
THEN MP_TAC (REAL_ARITH`~((v cross u) dot w = &0)==> &0< (v cross u) dot w \/ &0< --((v cross u) dot w)`)
THEN RESA_TAC
THENL[POP_ASSUM (fun th-> MP_TAC th THEN ONCE_REWRITE_TAC[CROSS_TRIPLE] THEN DISCH_TAC THEN POP_ASSUM(fun th-> MP_TAC th THEN ONCE_REWRITE_TAC[CROSS_TRIPLE] THEN DISCH_TAC THEN ASSUME_TAC th) THEN ASSUME_TAC th)
THEN MRESAL_TAC aff_gt_3_1_rep_cross_dot[`(vec 0):real^3`;`v:real^3`;`u:real^3`;`w:real^3`][VECTOR_ARITH`A- vec  0=A`;]
THEN MRESAL_TAC aff_gt_3_1_rep_cross_dot[`(vec 0):real^3`;`u:real^3`;`w:real^3`;`v:real^3`][VECTOR_ARITH`A- vec  0=A`;]
THEN MRESAL_TAC aff_gt_3_1_rep_cross_dot[`(vec 0):real^3`;`w:real^3`;`v:real^3`;`u:real^3`][VECTOR_ARITH`A- vec  0=A`;]
THEN MATCH_MP_TAC OPEN_INTER
THEN  REWRITE_TAC[REAL_ARITH`&0<A<=> A> &0`; OPEN_HALFSPACE_GT;] 
THEN MATCH_MP_TAC OPEN_INTER
THEN  REWRITE_TAC[ OPEN_HALFSPACE_GT;] ;

POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM DOT_LNEG]
THEN ONCE_REWRITE_TAC[GSYM CROSS_SKEW]
THEN ASM_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN REPEAT DISCH_TAC
THEN POP_ASSUM (fun th-> MP_TAC th THEN ONCE_REWRITE_TAC[CROSS_TRIPLE] THEN DISCH_TAC THEN POP_ASSUM(fun th-> MP_TAC th THEN ONCE_REWRITE_TAC[CROSS_TRIPLE] THEN DISCH_TAC THEN ASSUME_TAC th) THEN ASSUME_TAC th)
THEN MRESAL_TAC aff_gt_3_1_rep_cross_dot[`(vec 0):real^3`;`u:real^3`;`v:real^3`;`w:real^3`][VECTOR_ARITH`A- vec  0=A`;]
THEN MRESAL_TAC aff_gt_3_1_rep_cross_dot[`(vec 0):real^3`;`w:real^3`;`u:real^3`;`v:real^3`][VECTOR_ARITH`A- vec  0=A`;]
THEN MRESAL_TAC aff_gt_3_1_rep_cross_dot[`(vec 0):real^3`;`v:real^3`;`w:real^3`;`u:real^3`][VECTOR_ARITH`A- vec  0=A`;]
THEN MATCH_MP_TAC OPEN_INTER
THEN  REWRITE_TAC[REAL_ARITH`&0<A<=> A> &0`; OPEN_HALFSPACE_GT;] 
THEN MATCH_MP_TAC OPEN_INTER
THEN  REWRITE_TAC[ OPEN_HALFSPACE_GT;] ]);;


let OPEN_DIFF_AFF_GE=prove(`!x v u w:real^3.
open ((:real^3) DIFF (aff_ge {x} {v,u,w}))`,
REWRITE_TAC[OPEN_CLOSED;SET_RULE`(:real^3) DIFF ((:real^3) DIFF aff_ge {x} {v, u, w})= aff_ge {x} {v, u, w}`]
THEN REPEAT GEN_TAC
THEN MATCH_MP_TAC CLOSED_AFF_GE
THEN   SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY;
                 IN_INSERT; NOT_IN_EMPTY]);;


let aff_ge_1_3_eq_unions_aff_ge_1_2_and_aff_gt_1_3=prove(`!x  v u w:real^3.
~coplanar {x,v,u,w}
==>
aff_ge {x} {v,u,w}= aff_ge {x} {v,u} UNION aff_ge {x} {u,w} UNION aff_ge {x} {w,v} UNION  aff_gt {x} {v,u,w}`,
GEOM_ORIGIN_TAC `x:real^3`
THEN REPEAT STRIP_TAC
THEN MRESA_TAC notcoplanar_disjoints[`(vec 0):real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC notcoplanar_disjoint[`(vec 0):real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC AFF_GE_1_2[`(vec 0):real^3`;`v:real^3`;`u:real^3`]
THEN MRESA_TAC AFF_GE_1_2[`(vec 0):real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC AFF_GE_1_2[`(vec 0):real^3`;`w:real^3`;`v:real^3`]
THEN MRESA_TAC AFF_GT_1_3[`(vec 0):real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC AFF_GE_1_3[`(vec 0):real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC NOT_COPLANAR_0_4_IMP_INDEPENDENT[`v:real^3`;`u:real^3`;`w:real^3`]
THEN REWRITE_TAC[EXTENSION;IN_ELIM_THM;UNION]
THEN REDUCE_VECTOR_TAC
THEN GEN_TAC
THEN EQ_TAC
THENL[
STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0<= t2==> t2= &0 \/ &0< t2`) THEN RESA_TAC
THENL[

SUBGOAL_THEN ` (?t1 t2' t3'.
      &0 <= t2' /\
      &0 <= t3' /\
      t1 + t2' + t3' = &1 /\
      &0 % v + t3 % u + t4 % w = t2' % u + t3' % w:real^3)` ASSUME_TAC
THENL[
REDUCE_VECTOR_TAC
THEN EXISTS_TAC `t1:real`
THEN EXISTS_TAC `t3:real`
THEN EXISTS_TAC `t4:real`
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REAL_ARITH_TAC;
ASM_REWRITE_TAC[]];

MP_TAC(REAL_ARITH`&0<= t3==> t3= &0 \/ &0< t3`) THEN RESA_TAC
THENL[
SUBGOAL_THEN ` (?t1 t2' t3'.
      &0 <= t2' /\
      &0 <= t3' /\
      t1 + t2' + t3' = &1 /\
      t2 % v + &0 % u + t4 % w = t2' % w + t3' % v:real^3)` ASSUME_TAC
THENL[
REDUCE_VECTOR_TAC
THEN EXISTS_TAC `t1:real`
THEN EXISTS_TAC `t4:real`
THEN EXISTS_TAC `t2:real`
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t2 % v + t4 % w = t4 % w + t2 % v`]
THEN ASM_TAC
THEN REAL_ARITH_TAC;
ASM_REWRITE_TAC[]];

MP_TAC(REAL_ARITH`&0<= t4 ==> t4= &0 \/ &0< t4`) THEN RESA_TAC
THENL[
SUBGOAL_THEN ` (?t1 t2' t3'.
      &0 <= t2' /\
      &0 <= t3' /\
      t1 + t2' + t3' = &1 /\
      t2 % v + t3 % u + &0 % w = t2' % v + t3' % u:real^3)` ASSUME_TAC
THENL[
REDUCE_VECTOR_TAC
THEN EXISTS_TAC `t1:real`
THEN EXISTS_TAC `t2:real`
THEN EXISTS_TAC `t3:real`
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REAL_ARITH_TAC;
ASM_REWRITE_TAC[]];
SUBGOAL_THEN ` (?t1 t2' t3' t4'.
      &0 < t2' /\
      &0 < t3' /\
      &0 < t4' /\
      t1 + t2' + t3' + t4' = &1 /\
      t2 % v + t3 % u + t4 % w = t2' % v + t3' % u + t4' % w:real^3)` ASSUME_TAC
THENL[ REDUCE_VECTOR_TAC
THEN EXISTS_TAC `t1:real`
THEN EXISTS_TAC `t2:real`
THEN EXISTS_TAC `t3:real`
THEN EXISTS_TAC `t4:real`
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REAL_ARITH_TAC;
ASM_REWRITE_TAC[]]]]];

STRIP_TAC
THENL[
EXISTS_TAC `t1:real`
THEN EXISTS_TAC `t2:real`
THEN EXISTS_TAC `t3:real`
THEN EXISTS_TAC `&0:real`
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REAL_ARITH_TAC;

EXISTS_TAC `t1:real`
THEN EXISTS_TAC `&0:real`
THEN EXISTS_TAC `t2:real`
THEN EXISTS_TAC `t3:real`
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REAL_ARITH_TAC;

EXISTS_TAC `t1:real`
THEN EXISTS_TAC `t3:real`
THEN EXISTS_TAC `&0:real`
THEN EXISTS_TAC `t2:real`
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t2 % w + t3 % v = t3 % v + t2 % w:real^3`]
THEN ASM_TAC
THEN REAL_ARITH_TAC;

EXISTS_TAC `t1:real`
THEN EXISTS_TAC `t2:real`
THEN EXISTS_TAC `t3:real`
THEN EXISTS_TAC `t4:real`
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REAL_ARITH_TAC]]);;

	

let cut_aff_gt_1_3_connected=prove(`!x y z v u w:real^3 s:real^3->bool.
        connected s /\ ~coplanar {x,v,u,w} 
        /\  y IN s /\ z IN s /\ y IN aff_gt {x} {v,u,w} /\ ~(z IN aff_gt {x} {v,u,w})
        ==> ?t. t IN s /\ t IN aff_ge {x} {v,u} UNION aff_ge {x} {u,w} UNION aff_ge {x} {w,v}`,
  REPEAT STRIP_TAC THEN
  FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [connected]) THEN
  REWRITE_TAC[NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPECL
   [`aff_gt {x} {v,u,w:real^3}`; `(:real^3) DIFF (aff_ge {x} {v,u,w})`])
THEN MRESA_TAC OPEN_AFF_GT_1_3[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
 THEN REWRITE_TAC[ OPEN_DIFF_AFF_GE] THEN
  ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] 
THEN STRIP_TAC 
THEN
  REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_INTER; NOT_IN_EMPTY; SUBSET;
              IN_UNION; ] 
THEN MRESA_TAC aff_ge_1_3_eq_unions_aff_ge_1_2_and_aff_gt_1_3
[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM  MP_TAC
THEN  REWRITE_TAC[NOT_EXISTS_THM;UNION;DIFF;IN_ELIM_THM;DE_MORGAN_THM]
THEN DISCH_THEN (LABEL_TAC"A")
THEN STRIP_TAC
THENL[
REPEAT STRIP_TAC
THEN REMOVE_THEN "A"(fun th -> MRESA1_TAC th`x':real^3`)
THEN SET_TAC[];
STRIP_TAC
THENL[

REPEAT STRIP_TAC
THEN REMOVE_THEN "A"(fun th -> MRESA1_TAC th`x':real^3`)
THEN SET_TAC[];

STRIP_TAC
THENL[
ASM_MESON_TAC[];
REWRITE_TAC[NOT_FORALL_THM;DE_MORGAN_THM]
THEN EXISTS_TAC`z:real^3`
THEN REMOVE_THEN "A"(fun th -> MRESA1_TAC th`z:real^3`)
THEN SET_TAC[]]]]);;





let AFF_GE_SUBSET_XFAN=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) u:real^3 w:real^3.
 {u,w} IN E
==> 
aff_ge {x} {u,w} SUBSET xfan(x:real^3,V:real^3->bool,E)`,
ASM_REWRITE_TAC[xfan; SUBSET;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`{u,w:real^3}`
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN SET_TAC[IN]);;

let notcoplanar_4point_aff_gt_3_1_not_empty=prove(`!x v u w:real^3.
~coplanar {x,v,u,w}
==> ~(aff_gt {x,v,u} {w} INTER aff_gt {x,u,w} {v} INTER aff_gt {x,w,v} {u} = {})`,
REWRITE_TAC[SET_RULE`~(A={})<=> ?y. y IN A`]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM (fun th-> MP_TAC th THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,D,B}`] THEN DISCH_TAC  THEN
MP_TAC th THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,B,C}`] THEN DISCH_TAC THEN ASSUME_TAC th)
THEN MRESA_TAC coplanar_cross_dot[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MP_TAC(REAL_ARITH` ~(((v - x) cross (u - x)) dot (w - x) = &0)
==> &0 <((v - x) cross (u - x)) dot (w - x) \/  &0< --(((v - x) cross (u - x)) dot (w - x)) `)
THEN RESA_TAC
THENL[
POP_ASSUM (fun th-> MP_TAC th THEN ONCE_REWRITE_TAC[CROSS_TRIPLE] THEN DISCH_TAC THEN POP_ASSUM(fun th-> MP_TAC th THEN ONCE_REWRITE_TAC[CROSS_TRIPLE] THEN DISCH_TAC THEN ASSUME_TAC th) THEN ASSUME_TAC th)
THEN MRESA_TAC aff_gt_3_1_rep_cross_dot[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC aff_gt_3_1_rep_cross_dot[`x:real^3`;`u:real^3`;`w:real^3`;`v:real^3`]
THEN MRESAL_TAC aff_gt_3_1_rep_cross_dot[`x:real^3`;`w:real^3`;`v:real^3`;`u:real^3`][INTER; IN_ELIM_THM]
THEN EXISTS_TAC`&1/ &3 % v + &1/ &3 % u+ &1/ &3 % w:real^3`
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(&1 / &3 % v + &1 / &3 % u + &1 / &3 % w) - x= &1 / &3 % (v-x) + &1 / &3 % (u-x) + &1 / &3 % (w - x)`;DOT_RADD;DOT_RMUL;DOT_CROSS_SELF]
THEN REDUCE_ARITH_TAC
THEN REPEAT STRIP_TAC
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0< &1/ &3`] ;
POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM DOT_LNEG]
THEN ONCE_REWRITE_TAC[GSYM CROSS_SKEW]
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> MP_TAC th THEN ONCE_REWRITE_TAC[CROSS_TRIPLE] THEN DISCH_TAC THEN POP_ASSUM(fun th-> MP_TAC th THEN ONCE_REWRITE_TAC[CROSS_TRIPLE] THEN DISCH_TAC THEN ASSUME_TAC th) THEN ASSUME_TAC th)
THEN ASM_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN REPEAT STRIP_TAC
THEN MRESA_TAC aff_gt_3_1_rep_cross_dot[`x:real^3`;`u:real^3`;`v:real^3`;`w:real^3`]
THEN MRESA_TAC aff_gt_3_1_rep_cross_dot[`x:real^3`;`w:real^3`;`u:real^3`;`v:real^3`]
THEN MRESAL_TAC aff_gt_3_1_rep_cross_dot[`x:real^3`;`v:real^3`;`w:real^3`;`u:real^3`][INTER; IN_ELIM_THM]
THEN EXISTS_TAC`&1/ &3 % v + &1/ &3 % u+ &1/ &3 % w:real^3`
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(&1 / &3 % v + &1 / &3 % u + &1 / &3 % w) - x= &1 / &3 % (v-x) + &1 / &3 % (u-x) + &1 / &3 % (w - x)`;DOT_RADD;DOT_RMUL;DOT_CROSS_SELF]
THEN REDUCE_ARITH_TAC
THEN REPEAT STRIP_TAC
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0< &1/ &3`]]) ;;

let notcoplanar_4point_aff_gt_1_3_not_empty=prove(`!x v u w:real^3.
~coplanar {x,v,u,w}
==> ~(aff_gt {x} {v,u,w}  = {})`,
REPEAT GEN_TAC THEN STRIP_TAC
THEN MRESA_TAC inter_aff_gt_3_1_is_aff_gt_1_3[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM th])
THEN POP_ASSUM MP_TAC
THEN SET_TAC[notcoplanar_4point_aff_gt_3_1_not_empty]);;

let KVQWYDL_lemma1=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ {w,v} IN E
/\ sigma_fan x V E u w = v
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
==>  aff_gt{x} {v,u,w} = dart_leads_into x V E u w`,
REPEAT STRIP_TAC
THEN POP_ASSUM (fun th -> MP_TAC th
THEN REWRITE_TAC[fan80] THEN ASSUME_TAC th)
THEN DISCH_TAC
THEN POP_ASSUM (fun th -> MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC "YEU EM"))
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MATCH_MP_TAC (SET_RULE`A SUBSET B /\ B SUBSET A==> A=B`)
THEN MRESA_TAC aff_gt_1_3_subset_dart_leads_into_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` v:real^3`;` u:real^3`;` w:real^3`]
THEN MATCH_MP_TAC (SET_RULE`~(?z. z IN A /\ ~(z IN B))==> A SUBSET B`)
THEN STRIP_TAC
THEN MRESA_TAC connected_dart_leads_into_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` u:real^3`;` w:real^3`]
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC notcoplanar_4point_aff_gt_1_3_not_empty[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`][SET_RULE`~(A={})<=> ?y. y IN A`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`aff_gt {x} {v, u, w} SUBSET dart_leads_into x V E u w /\ y IN aff_gt {x} {v, u, w}
==>  y IN dart_leads_into x V E u w `)
THEN RESA_TAC
THEN MRESA_TAC cut_aff_gt_1_3_connected[`x:real^3`;`y:real^3`;`z:real^3`;`v:real^3`;`u:real^3`;`w:real^3`;`(dart_leads_into x V E u w):real^3->bool`]
THEN MRESA_TAC dart_leads_into_fan_in_topological_component_yfan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC topological_component_subset_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (dart_leads_into x V E u w):real^3->bool`]
THEN MP_TAC (SET_RULE`t IN dart_leads_into x V E u w /\ dart_leads_into x V E u w SUBSET yfan (x,V,E)
==>t IN  yfan (x,V,E)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[yfan]
THEN MRESA_TAC AFF_GE_SUBSET_XFAN[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`]
THEN MRESA_TAC AFF_GE_SUBSET_XFAN[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC AFF_GE_SUBSET_XFAN[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`w:real^3`;`v:real^3`]
THEN ASM_TAC
THEN SET_TAC[]);;




let point_in_yfan_is_not_inv_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool z:real^3 u:real^3.
FAN(x,V,E) /\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ U IN topological_component_yfan (x,V,E)
/\ z IN U
/\ u IN V
==> ~(u=z)`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC nonsetedge_fully_surround_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`]
THEN MRESA_TAC zpoint_in_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`U:real^3->bool`; `z:real^3`;]
THEN MRESA_TAC v_subset_xfan[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`]
THEN ASM_TAC THEN REWRITE_TAC[yfan]
THEN SET_TAC[]);;



let point_in_aff_ge_1_1=prove(`!(x:real^3) (v:real^3).
~(x=v)
==>
x IN aff_ge {x} {v}
/\ v IN aff_ge {x} {v}`,

REPEAT GEN_TAC THEN STRIP_TAC 
  THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`]AFF_GE_1_1)
THEN RESA_TAC
		THEN ASM_REWRITE_TAC[IN_ELIM_THM]
THEN STRIP_TAC
THENL[ EXISTS_TAC`&1:real`
THEN EXISTS_TAC`&0:real`
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC;  

 EXISTS_TAC`&0:real`
THEN EXISTS_TAC`&1:real`
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC]);;


let POINT_IN_AFF_GE_IMP_IN_EDGE=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 u1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ u1 IN V /\ ~(x=u1)
/\ u1 IN aff_ge {x} {v,u}
==> u1 IN {v,u}`,

REPEAT STRIP_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC aff_ge_1_1_subset_aff_ge_fan[`x:real^3`;`v:real^3`;`u:real^3`;`u1:real^3`]
THEN MP_TAC(SET_RULE`aff_ge {x} {u1} SUBSET aff_ge {x} {v, u}==>
aff_ge {x} {u1} INTER aff_ge {x} {v, u:real^3}=aff_ge {x} {u1}`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`?v. v IN V /\ {u1} = {v:real^3} <=> u1 IN V`)
THEN ASM_REWRITE_TAC[]
THEN DISCH_TAC
THEN FIND_ASSUM MP_TAC `FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN;fan7]
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> MRESAL_TAC th[`{u1:real^3}`;`{v,u:real^3}`][UNION;IN_ELIM_THM;])
THEN POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`{u1} INTER {v,u:real^3}= {}\/ u1 IN {v, u}`)
THENL [ASM_REWRITE_TAC[AFF_GE_EQ_AFFINE_HULL;AFFINE_HULL_1;SET_RULE`{u % x| u= &1} ={&1 %x}`; VECTOR_ARITH`&1 %X=X`]
THEN STRIP_TAC
THEN MRESAL_TAC point_in_aff_ge_1_1[`(x:real^3)`;`(u1:real^3) `][IN_SING];
ASM_REWRITE_TAC[]]);;






let POINT_IN_CLOSURE_AFF_GT_1_2=prove(`!x:real^3 v:real^3 u:real^3.
~(x=v) /\ ~(x=u) /\ ~(v=u) 
==> v IN closure(aff_gt {x} {v,u})`,
REPEAT GEN_TAC
THEN GEOM_ORIGIN_TAC `x:real^3`
THEN REPEAT STRIP_TAC
THEN MP_TAC(SET_RULE`~((vec 0:real^3) =v) /\ ~((vec 0:real^3)=u) /\ ~(v=u:real^3) ==> DISJOINT {vec 0} {v,u}/\ ~(u=v)`) THEN RESA_TAC
THEN MRESAL_TAC AFF_GT_1_2[`vec 0:real^3`;`v:real^3`;`u:real^3`][CLOSURE_APPROACHABLE;IN_ELIM_THM;dist]
THEN REDUCE_VECTOR_TAC
THEN REPEAT STRIP_TAC
THEN ABBREV_TAC`t3=min (inv(norm(u-v:real^3)) * e) (&1)/ &2:real`
THEN ABBREV_TAC`t2= &1- t3:real`
THEN MRESA_TAC IMP_NORM_FAN[`u:real^3`;`v:real^3`] 
THEN MRESA_TAC REAL_LT_MUL[`inv (norm (u - v:real^3))`;`e:real`]
THEN MP_TAC(REAL_ARITH`&0 < inv (norm (u - v:real^3)) * e /\ t3=min (inv(norm(u-v:real^3)) * e) (&1)/ &2
==> &0<= t3/\ &0< t3 /\ t3< &1 /\ &0< &1- t3 /\ t3< inv (norm (u - v:real^3)) * e `) THEN RESA_TAC
THEN MRESA1_TAC REAL_ABS_REFL`t3:real`
THEN MRESAL_TAC REAL_LT_LMUL[`norm (u - v:real^3)`;`t3:real`;`inv (norm (u - v:real^3)) * e:real`][REAL_ARITH`A*B*C=(B*A)*C`]
THEN EXISTS_TAC`t2%v +t3 % u:real^3`
THEN STRIP_TAC
THENL[
EXISTS_TAC`&0`
THEN EXISTS_TAC `t2:real`
THEN EXISTS_TAC `t3:real`
THEN ASM_REWRITE_TAC[]
THEN EXPAND_TAC "t2"
THEN REAL_ARITH_TAC;
EXPAND_TAC "t2"
THEN REWRITE_TAC[VECTOR_ARITH`((&1 - t3) % v + t3 % u) - v=t3% (u-v)`;NORM_MUL]
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC]);;



let KVQWYDL_lemma2=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 u1:real^3 w1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ {w,v} IN E /\ {u1,w1} IN E
/\ sigma_fan x V E u w = v
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\   aff_gt{x} {v,u,w} = dart_leads_into x V E u1 w1
==> u1 IN {v,u,w}`,


REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th -> MP_TAC th
THEN REWRITE_TAC[fan80] THEN ASSUME_TAC th)
THEN DISCH_TAC
THEN POP_ASSUM (fun th -> MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC "YEU EM"))
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC not_empty_rw_dart_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`u1:real^3`;`w1:real^3`]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"EM")
THEN MRESA_TAC DART_LEADS_INTO[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;
`u1:real^3`;`w1:real^3`;]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"YEU")
THEN MRESA_TAC rw_dart_avoids_fan[`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`u1:real^3`;`w1:real^3`]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"OI")
THEN MP_TAC(REAL_ARITH`&1> h' /\ h' > &0==> -- &1 < h' /\ h'< &1 /\ -- &1 <= h' /\ h'<= &1/\ &0 < h' /\ h' <= &1`) THEN RESA_TAC
THEN MRESA1_TAC ACS_BOUNDS_LT`h':real`
THEN MRESAL_TAC ACS_MONO_LT[`&0`;`h':real`][ACS_0;REAL_ARITH`-- &1 <= &0`]
THEN MRESA1_TAC COS_ACS `h':real`
THEN ABBREV_TAC`h1= min (h:real) (acs h')/ &2`
THEN MP_TAC(REAL_ARITH`h1= min (h:real) (acs h')/ &2 /\ &0<h /\ &0< acs h' /\ acs h'< pi/ &2==> &0< h1 /\ h1< h /\ h1<pi/ &2/\ h1< acs h' /\ acs h' <= pi /\ &0<= h1`)
THEN ASM_REWRITE_TAC[PI_WORKS] THEN STRIP_TAC
THEN MRESAL_TAC COS_MONO_LT[`h1:real`;`acs h':real`][ACS_0;REAL_ARITH`-- &1 <= &0`]
THEN MP_TAC(REAL_ARITH`h'< cos h1==> h'<= cos h1`) THEN RESA_TAC
THEN REMOVE_THEN "EM"(fun th-> MRESAL1_TAC th `h1:real`[SET_RULE`~(A={})<=> ?y. y IN A`])
THEN REMOVE_THEN "YEU"(fun th-> MRESA_TAC th [`h1:real`;`y:real^3`])
THEN MRESA_TAC sigma_fan_in_set_of_edge[`x:real^3`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`u1:real^3`;`w1:real^3`]
THEN POP_ASSUM MP_TAC 
THEN MRESA_TAC properties_of_set_of_edge_fan [`x:real^3`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`u1:real^3`;`w1:real^3`]
THEN DISCH_TAC
THEN MRESA_TAC properties_of_set_of_edge_fan [`x:real^3`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`u1:real^3`;`sigma_fan x V E u1 w1:real^3`]
THEN MRESA_TAC aff_gt_in_rw_dart_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`sigma_fan x V E u1 w1:real^3`;`u1:real^3`;`w1:real^3`;`y:real^3`;`h1:real`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN STRIP_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w1:real^3)`;
` (u1:real^3)`]
THEN MRESA_TAC dart_leads_into_fan_in_topological_component_yfan[`x:real^3`;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`u1:real^3`;`w1:real^3`]
THEN MP_TAC (SET_RULE`y IN  rw_dart_fan x V E (x,u1,w1,sigma_fan x V E u1 w1) (cos h1)
/\ rw_dart_fan x V E (x,u1,w1,sigma_fan x V E u1 w1) (cos h1) SUBSET
      dart_leads_into x V E u1 w1
==> y IN dart_leads_into x V E u1 w1`)
THEN RESA_TAC
THEN MRESA_TAC point_in_yfan_is_not_inv_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`; `(dart_leads_into x V E u1 w1):real^3->bool`;`y:real^3`;`u1:real^3`]
THEN MRESA_TAC nonsetedge_fully_surround_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`]
THEN MRESA_TAC point_in_yfan_not_x_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(dart_leads_into x V E u1 w1):real^3->bool`;` y:real^3`]
THEN DISCH_THEN(fun th-> ASSUME_TAC (SYM th))
THEN MP_TAC (SET_RULE`aff_gt {x} {u1, y} SUBSET
      rw_dart_fan x V E (x,u1,w1,sigma_fan x V E u1 w1) (cos h1)
/\ aff_gt {x} {v, u, w} = dart_leads_into x V E u1 w1
/\ rw_dart_fan x V E (x,u1,w1,sigma_fan x V E u1 w1) (cos h1) SUBSET
      dart_leads_into x V E u1 w1
==> aff_gt {x} {u1, y} SUBSET aff_gt {x} {v, u, w}`)
THEN RESA_TAC
THEN MRESA_TAC aff_ge_1_3_eq_unions_aff_ge_1_2_and_aff_gt_1_3[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MP_TAC (SET_RULE`aff_gt {x} {u1, y} SUBSET aff_gt {x} {v, u, w}
/\ aff_ge {x} {v, u, w} =
      aff_ge {x} {v, u} UNION
      aff_ge {x} {u, w} UNION
      aff_ge {x} {w, v} UNION
      aff_gt {x} {v, u, w:real^3}
==> aff_gt {x} {u1, y} SUBSET aff_ge {x} {v, u, w}`)
THEN POP_ASSUM(fun th-> GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN RESA_TAC
THEN SUBGOAL_THEN`closed (aff_ge {x:real^3} {v, u, w})` ASSUME_TAC
THENL[MATCH_MP_TAC	CLOSED_AFF_GE
THEN SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY;
                 IN_INSERT; NOT_IN_EMPTY];
MRESA1_TAC CLOSURE_EQ`aff_ge {x:real^3} {v, u, w}`
THEN MRESA_TAC SUBSET_CLOSURE[`aff_gt {x} {u1, y:real^3}`;`aff_ge {x:real^3} {v, u, w}`]
THEN MRESA_TAC POINT_IN_CLOSURE_AFF_GT_1_2[`x:real^3`;`u1:real^3`;`y:real^3`]
THEN MP_TAC (SET_RULE`u1 IN closure(aff_gt {x} {u1, y})
/\ closure (aff_gt {x} {u1, y}) SUBSET aff_ge {x} {v, u, w}
==> u1 IN aff_ge {x} {v, u, w:real^3}`)
THEN ASM_REWRITE_TAC[]
 THEN MRESA_TAC aff_ge_1_3_eq_unions_aff_ge_1_2_and_aff_gt_1_3[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESAL_TAC topological_component_subset_yfan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` (dart_leads_into x V E u1 w1):real^3->bool`][yfan]
THEN MRESA_TAC v_subset_xfan[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`aff_gt {x} {v, u, w} SUBSET (:real^3) DIFF xfan (x,V,E) /\
 V SUBSET xfan (x,V,E)
/\ u1 IN V
/\ u1 IN
 aff_ge {x} {v, u} UNION
 aff_ge {x} {u, w} UNION
 aff_ge {x} {w, v} UNION
 aff_gt {x} {v, u, w}
==> u1 IN
 aff_ge {x} {v, u} UNION
 aff_ge {x} {u, w} UNION
 aff_ge {x} {w, v} `)
THEN ASM_REWRITE_TAC[UNION;IN_ELIM_THM]
THEN STRIP_TAC
THENL[
MRESA_TAC POINT_IN_AFF_GE_IMP_IN_EDGE[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`; `u:real^3`;`u1:real^3`]
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];MRESA_TAC POINT_IN_AFF_GE_IMP_IN_EDGE[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`u:real^3`; `w:real^3`;`u1:real^3`]
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];
MRESA_TAC POINT_IN_AFF_GE_IMP_IN_EDGE[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`w:real^3`; `v:real^3`;`u1:real^3`]
THEN POP_ASSUM MP_TAC
THEN SET_TAC[]]]);;



let DISJOINT_RW_DART_FAN_SAME_NODE_FAN=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 h1:real h2:real.
FAN(x,V,E)/\ {u,w} IN E /\ {u,w1} IN E/\ ~(w=w1)
==> rw_dart_fan x V E (x,u,w,sigma_fan x V E u w) (cos h1)
INTER rw_dart_fan x V E (x,u,w1,sigma_fan x V E u w1) (cos h2)= {}`,
REPEAT STRIP_TAC
THEN MRESA_TAC disjoint_fan2[`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(u:real^3)`;`(w:real^3)`;` (w1:real^3)`]
THEN REWRITE_TAC[rw_dart_fan]
THEN POP_ASSUM MP_TAC
THEN SET_TAC[]);;



let condition_unique_by_dart_leads_into=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) w:real^3 u:real^3 w1:real^3.
FAN(x,V,E)/\ {u,w} IN E /\ {u,w1} IN E
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ dart_leads_into x V E u w SUBSET w_dart_fan x V E (x,u,w,sigma_fan x V E u w)
/\  dart_leads_into x V E u w  = dart_leads_into x V E u w1 
==> w= w1`,

REPEAT STRIP_TAC
THEN MRESA_TAC not_empty_rw_dart_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`u:real^3`;`w1:real^3`]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"EM")
THEN MRESA_TAC DART_LEADS_INTO[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;
`u:real^3`;`w1:real^3`;]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"YEU")
THEN MRESA_TAC rw_dart_avoids_fan[`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`u:real^3`;`w1:real^3`]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"OI")
THEN MP_TAC(REAL_ARITH`&1> h' /\ h' > &0==> -- &1 < h' /\ h'< &1 /\ -- &1 <= h' /\ h'<= &1/\ &0 < h' /\ h' <= &1`) THEN RESA_TAC
THEN MRESA1_TAC ACS_BOUNDS_LT`h':real`
THEN MRESAL_TAC ACS_MONO_LT[`&0`;`h':real`][ACS_0;REAL_ARITH`-- &1 <= &0`]
THEN MRESA1_TAC COS_ACS `h':real`
THEN ABBREV_TAC`h1= min (h:real) (acs h')/ &2`
THEN MP_TAC(REAL_ARITH`h1= min (h:real) (acs h')/ &2 /\ &0<h /\ &0< acs h' /\ acs h'< pi/ &2==> &0< h1 /\ h1< h /\ h1<pi/ &2/\ h1< acs h' /\ acs h' <= pi /\ &0<= h1`)
THEN ASM_REWRITE_TAC[PI_WORKS] THEN STRIP_TAC
THEN MRESAL_TAC COS_MONO_LT[`h1:real`;`acs h':real`][ACS_0;REAL_ARITH`-- &1 <= &0`]
THEN MP_TAC(REAL_ARITH`h'< cos h1==> h'<= cos h1`) THEN RESA_TAC
THEN REMOVE_THEN "EM"(fun th-> MRESAL1_TAC th `h1:real`[SET_RULE`~(A={})<=> ?y. y IN A`])
THEN POP_ASSUM (fun th-> MP_TAC th THEN REWRITE_TAC[rw_dart_fan] THEN STRIP_TAC THEN ASSUME_TAC th)
THEN REMOVE_THEN "YEU"(fun th-> MRESA_TAC th [`h1:real`;`y:real^3`])
THEN MP_TAC(SET_RULE` dart_leads_into x V E u w SUBSET w_dart_fan x V E (x,u,w,sigma_fan x V E u w)
/\  dart_leads_into x V E u w  = dart_leads_into x V E u w1 
/\ rw_dart_fan x V E (x,u,w1,sigma_fan x V E u w1) (cos h1) SUBSET
      dart_leads_into x V E u w1
/\ y IN rw_dart_fan x V E (x,u,w1,sigma_fan x V E u w1) (cos h1)
/\ y IN
      w_dart_fan x V E (x,u,w1,sigma_fan x V E u w1) INTER
      rcone_fan x u (cos h1)
==> ~(w_dart_fan x V E (x,u,w,sigma_fan x V E u w) INTER w_dart_fan x V E (x,u,w1,sigma_fan x V E u w1) ={})`)
THEN RESA_TAC
THEN DISJ_CASES_TAC(SET_RULE`~(w=w1:real^3)\/ w=w1`)
THENL[
MRESA_TAC disjoint_fan2[`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(u:real^3)`;`(w:real^3)`;` (w1:real^3)`];
ASM_REWRITE_TAC[]]);;


let PROPERTIES_TRIANGLE_FAN_lemma1=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ {w,v} IN E 
/\ sigma_fan x V E u w = v
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
==> sigma_fan x V E v u = w `,

REPEAT STRIP_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (u:real^3)`]  
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (w:real^3)`;
` (v:real^3)`]  
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN REPEAT STRIP_TAC
THEN MP_TAC(SET_RULE`u IN set_of_edge v V E /\ w IN set_of_edge v V E /\ ~(u=w)==> ~( set_of_edge v V E= {u:real^3}) `)
THEN RESA_TAC
THEN  MATCH_MP_TAC UNIQUE_SIGMA_FAN
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MRESA_TAC SIGMA_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `w1:real^3`)
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC angle_is_small_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`; `w:real^3`]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC);;

let PROPERTIES_TRIANGLE_FAN_lemma2=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ {w,v} IN E 
/\ sigma_fan x V E u w = v
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
==> sigma_fan x V E w v=u`,
REPEAT STRIP_TAC
THEN MRESA_TAC PROPERTIES_TRIANGLE_FAN_lemma1[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`; `w:real^3`]
THEN MRESA_TAC PROPERTIES_TRIANGLE_FAN_lemma1[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`w:real^3`;`v:real^3`; `u:real^3`]);;

let PROPERTIES_TRIANGLE_FAN=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ {w,v} IN E 
/\ sigma_fan x V E u w = v
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
==> sigma_fan x V E v u = w/\ sigma_fan x V E w v=u`,
MESON_TAC[PROPERTIES_TRIANGLE_FAN_lemma1; PROPERTIES_TRIANGLE_FAN_lemma2]);;




let inter_aff_gt_3_1_is_aff_gt_2_2=prove(`!x v u w:real^3.
~coplanar {x,v,u,w}
==>
aff_gt {x,v,u} {w} INTER aff_gt {x,u,w} {v} =aff_gt {x,u} {v,w}`,
GEOM_ORIGIN_TAC `x:real^3`
THEN REPEAT STRIP_TAC
THEN MRESA_TAC notcoplanar_disjoints[`(vec 0):real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC notcoplanar_disjoint[`(vec 0):real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC AFF_GT_3_1[`(vec 0):real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC AFF_GT_3_1[`(vec 0):real^3`;`u:real^3`;`w:real^3`;`v:real^3`]
THEN MRESA_TAC AFF_GT_2_2[`(vec 0):real^3`;`u:real^3`;`v:real^3`;`w:real^3`]
THEN MRESA_TAC NOT_COPLANAR_0_4_IMP_INDEPENDENT[`v:real^3`;`u:real^3`;`w:real^3`]
THEN REWRITE_TAC[EXTENSION;IN_ELIM_THM;INTER]
THEN REDUCE_VECTOR_TAC
THEN GEN_TAC
THEN EQ_TAC
THENL[
STRIP_TAC
THEN POP_ASSUM MP_TAC 
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t2 % v + t3 % u + t4 % w = t2' % u + t3' % w + t4' % v <=> (t2-t4') % v + (t3-t2') % u + (t4-t3') % w= vec 0`]
THEN STRIP_TAC
THEN MRESAL_TAC INDEPENDENT_3[`v:real^3`;`u:real^3`;`w:real^3`;`t2-t4':real`; `t3-t2':real`; `t4 -t3':real`][REAL_ARITH`A-B= &0<=> A=B`]
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`t1':real`
THEN EXISTS_TAC`t2':real`
THEN EXISTS_TAC`t4':real`
THEN EXISTS_TAC`t3':real`
THEN ASM_REWRITE_TAC[REAL_ARITH`t1+t2+t3+t4=t1+t2+t4+t3:real`;VECTOR_ARITH`t4' % v + t2' % u + t3' % w = t2' % u + t4' % v + t3' % w:real^3`]
THEN ASM_TAC
THEN REAL_ARITH_TAC;
STRIP_TAC
THEN STRIP_TAC
THENL[
EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t3:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t4:real`
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t2 % u + t3 % v + t4 % w = t3 % v + t2 % u + t4 % w:real^3`;REAL_ARITH`t1 + t3 + t2 + t4=t1 + t2 + t3 + t4`];
EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t4:real`
THEN EXISTS_TAC`t3:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`t1 + t2 + t4+ t3:real=t1 + t2 + t3 + t4`]
THEN VECTOR_ARITH_TAC]]);;





let condition_edge_in_face_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 u1:real^3 w1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ {w,v} IN E /\ {u1,w1} IN E
/\ sigma_fan x V E u w = v
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ u1=v
/\   aff_gt{x} {v,u,w} = dart_leads_into x V E u1 w1
==> (u1,w1) IN {(v,u),(u,w),(w,v)}`,


REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC "EM")
THEN DISCH_THEN (LABEL_TAC "YEU")
THEN SUBGOAL_THEN`u=w1:real^3`ASSUME_TAC
THENL[
MATCH_MP_TAC condition_unique_by_dart_leads_into
THEN EXISTS_TAC `x:real^3`
THEN EXISTS_TAC `V:real^3->bool`
THEN EXISTS_TAC `E:(real^3->bool)->bool`
THEN EXISTS_TAC `v:real^3`
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THENL[ REMOVE_THEN "EM"(fun th-> ASM_REWRITE_TAC[SYM th]);
 STRIP_TAC
THENL[

MRESA_TAC PROPERTIES_TRIANGLE_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;` w:real^3`]
THEN MRESA_TAC KVQWYDL_lemma1[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`w:real^3`;`v:real^3`; `u:real^3`]
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM th])
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (u:real^3)`;
` (v:real^3)`]
THEN FIND_ASSUM MP_TAC`(!v:real^3. v IN V==>CARD (set_of_edge v V E) >1)` 
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> MP_TAC(ISPEC `v:real^3` th) THEN FIND_ASSUM MP_TAC `(v:real^3) IN V` THEN DISCH_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[th]) THEN STRIP_TAC)
THEN ASM_REWRITE_TAC[w_dart_fan]
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` (v:real^3)`;
` (w:real^3)`]
THEN FIND_ASSUM MP_TAC`fan80 (x:real^3,V,E)`
THEN DISCH_TAC
THEN POP_ASSUM (fun th -> MP_TAC th
THEN REWRITE_TAC[fan80] THEN ASSUME_TAC th)
THEN DISCH_TAC
THEN POP_ASSUM (fun th -> MRESA_TAC th [`v:real^3`;`u:real^3`] THEN MRESA_TAC th [`u:real^3`;`w:real^3`] THEN MP_TAC th THEN DISCH_THEN(LABEL_TAC "YEU EM"))
THEN MRESA_TAC WEDGE_LUNE_GT[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN RESA_TAC
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM (fun th-> MP_TAC th THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,C,B,D}`] THEN DISCH_TAC THEN MP_TAC th THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,C,B}`] THEN DISCH_TAC THEN ASSUME_TAC th)
THEN MRESA_TAC inter_aff_gt_3_1_is_aff_gt_2_2[`x:real^3`;`u:real^3`;`v:real^3`;`w:real^3`]
THEN MRESA_TAC inter_aff_gt_3_1_is_aff_gt_1_3[`x:real^3`;`w:real^3`;`u:real^3`;`v:real^3`]
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM th])
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM th])
THEN SET_TAC[];
REMOVE_THEN "YEU" MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th] THEN ASSUME_TAC th)
THEN DISCH_TAC 
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC PROPERTIES_TRIANGLE_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;` w:real^3`]
THEN MRESA_TAC KVQWYDL_lemma1[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`w:real^3`;`v:real^3`; `u:real^3`]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={C,A,B}`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])]];

ASM_REWRITE_TAC[]
THEN SET_TAC[]]);;

let KVQWYDL_lemma3=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 w:real^3 u1 w1.
FAN(x,V,E)/\ {v,u} IN E /\ {u,w} IN E /\ {w,v} IN E /\ {u1,w1} IN E
/\ sigma_fan x V E u w = v
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\   aff_gt{x} {v,u,w} = dart_leads_into x V E u1 w1
==> (u1,w1) IN {(v,u),(u,w),(w,v)}`,

REPEAT STRIP_TAC
THEN POP_ASSUM (fun th-> MP_TAC th THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,C,A}`] THEN DISCH_TAC
THEN MP_TAC th THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={C,A,B}`] THEN DISCH_TAC
THEN ASSUME_TAC th)
THEN MRESA_TAC KVQWYDL_lemma2 [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;`w:real^3`;`u1:real^3`;` w1:real^3`]
THEN MRESA_TAC PROPERTIES_TRIANGLE_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`;` w:real^3`]
THEN MP_TAC(SET_RULE`u1 IN {v, u, w}==> u1=v \/ u1= u \/ u1=w:real^3`)
THEN RESA_TAC
THENL[
MRESA_TAC  condition_edge_in_face_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`v:real^3`; `u:real^3`;`w:real^3`;`u1:real^3`;`w1:real^3`];
MRESA_TAC  condition_edge_in_face_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`; `u:real^3`;`w:real^3`;`v:real^3`;`u1:real^3`;`w1:real^3`]
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];
MRESA_TAC  condition_edge_in_face_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`; `w:real^3`;`v:real^3`;`u:real^3`;`u1:real^3`;`w1:real^3`]
THEN POP_ASSUM MP_TAC
THEN SET_TAC[]]);;




let CARD_3=prove(`!e:A->bool.
 CARD e=3 /\ FINITE e
==> ?v u w:A. e={v,u,w} /\ ~(v=u)/\ ~(u=w) /\ ~(w=v)`,
REPEAT GEN_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN `~((e:A->bool)={})` ASSUME_TAC
THENL[
STRIP_TAC
THEN MP_TAC(ISPEC`(e:A->bool)`CARD_EQ_0)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC 
THEN SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY;
                 IN_INSERT; NOT_IN_EMPTY]
THEN ARITH_TAC;

MP_TAC(SET_RULE `~((e:A->bool)={})==> ?v:A. v IN e`)
THEN RESA_TAC
THEN SUBGOAL_THEN`~((e:A->bool) DELETE v={})` ASSUME_TAC
THENL[
STRIP_TAC
THEN MP_TAC(ISPECL[`v:A`;`(e:A->bool)`;]CARD_DELETE)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(e:A->bool)`;`v:A`] FINITE_DELETE)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(e:A->bool) DELETE v`CARD_EQ_0)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
MP_TAC(SET_RULE `~((e:A->bool)DELETE v={})==> ?w:A. w IN (e:A->bool)DELETE v/\ w IN e`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE `(w IN (e:A->bool)DELETE v)==> ~(v=w)`)
THEN RESA_TAC
THEN SUBGOAL_THEN`~((e:A->bool) DELETE v DELETE w ={})` ASSUME_TAC
THENL[
STRIP_TAC
THEN MP_TAC(ISPECL[`v:A`;`(e:A->bool)`;]CARD_DELETE)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(e:A->bool)`;`v:A`] FINITE_DELETE)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`w:A`;`(e:A->bool) DELETE v`;]CARD_DELETE)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(e:A->bool) DELETE v`;`w:A`] FINITE_DELETE)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(e:A->bool) DELETE v DELETE w`CARD_EQ_0)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
MP_TAC(SET_RULE `~((e:A->bool) DELETE v DELETE w={})==> ?u:A. u IN ((e:A->bool)DELETE v DELETE w)/\ u IN e`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE `(v IN (e:A->bool))/\ (w IN (e:A->bool)) /\ (u IN (e:A->bool))==> {v,w,u} SUBSET e`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE `(u IN (e:A->bool)DELETE v DELETE w)==> ~(u=v)/\ ~(w=u)`)
THEN RESA_TAC
THEN SUBGOAL_THEN(`CARD {v:A,w,u}=3`)ASSUME_TAC
THENL[
SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY;
                 IN_INSERT; NOT_IN_EMPTY]

THEN ASM_REWRITE_TAC[]
THEN ARITH_TAC;


POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`CARD (e:A->bool)=3`
THEN DISCH_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)])
THEN DISCH_TAC
THEN MP_TAC(ISPECL[`{v:A,w,u}`;`e:A->bool`]CARD_SUBSET_EQ)
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)])
THEN RESA_TAC
THEN EXISTS_TAC`v:A`
THEN EXISTS_TAC`w:A`
THEN EXISTS_TAC`u:A`
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)])
THEN ASM_TAC
THEN SET_TAC[]]]]]);; 


let FINITE_FACE_FAN=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds.
FAN(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
==> FINITE ds `,
REPEAT STRIP_TAC
THEN  MRESA_TAC face_subset_dart_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN MRESA_TAC finite_d_fan[`x:real^3`;`V:real^3->bool`;`E:(real^3->bool)->bool`]
THEN MRESA_TAC FINITE_SUBSET[`ds:real^3#real^3#real^3#real^3->bool`;`d_fan (x:real^3, V:real^3->bool, (E:(real^3->bool)->bool))`]);;


let condition_f1_fan_in_face_set=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) y y1 ds.
FAN(x,V,E) 
/\ y = f1_fan x V E y1 
/\ ds IN face_set (hypermap1_of_fanx (x,V,E))
/\ d_fan (x,V,E) =d1_fan (x,V,E)
/\ y1 IN ds
==> y IN ds`,
REPEAT GEN_TAC THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC hypermap_of_fan_rep[`x:real^3`;`V:real^3->bool`;`(E:(real^3->bool)->bool)`;`(\t. res (t x V E) (d1_fan (x,V,E)))`]
THEN MRESA_TAC lemma_face_representation[`hypermap1_of_fanx(x:real^3,V,E)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN ASM_REWRITE_TAC[face; orbit_map;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC into_domain_power_efn_fan[`x:real^3`;`V:real^3->bool`;`(E:(real^3->bool)->bool)`;`n:num`;`( \ t. res (t x V E ) (d1_fan (x:real^3, V:real^3->bool, (E:(real^3->bool)->bool))))`]

THEN POP_ASSUM (fun th-> MRESA1_TAC th `x':real^3#real^3#real^3#real^3`)
THEN STRIP_TAC
THEN EXISTS_TAC`SUC n:num`
THEN MRESA_TAC into_domain_power_efn_fan[`x:real^3`;`V:real^3->bool`;`(E:(real^3->bool)->bool)`;`SUC n:num`;`( \ t. res (t x V E ) (d1_fan (x:real^3, V:real^3->bool, (E:(real^3->bool)->bool))))`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `x':real^3#real^3#real^3#real^3`)
THEN REWRITE_TAC[COM_POWER;o_DEF]
THEN ASM_TAC
THEN ARITH_TAC);;


let CARD_FACE_SET_GE_3_FULLY_SURROUNDED_FAN=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
==> 3<= CARD ds`,

REPEAT STRIP_TAC
THEN MRESA_TAC hypermap_of_fan_rep[`x:real^3`;`V:real^3->bool`;`(E:(real^3->bool)->bool)`;`(\t. res (t x V E) (d1_fan (x,V,E)))`]
THEN MRESA_TAC dartset_fully_surrounded_is_non_isolated_fan[`x:real^3`;`(V:real^3->bool)`; `(E:(real^3->bool)->bool)`]
THEN MRESA_TAC lemma_face_representation[`hypermap1_of_fanx(x:real^3,V,E)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN SUBGOAL_THEN`x' IN ds:real^3#real^3#real^3#real^3->bool` ASSUME_TAC

THENL (*1*)[
ASM_REWRITE_TAC[face;orbit_map;IN_ELIM_THM;]
THEN EXISTS_TAC`0`
THEN REWRITE_TAC[POWER;I_DEF]
THEN ARITH_TAC;(*1*)
ABBREV_TAC`y=f1_fan x V E (x':real^3#real^3#real^3#real^3)`
THEN MRESA_TAC condition_f1_fan_in_face_set[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(y:real^3#real^3#real^3#real^3)` ;`(x':real^3#real^3#real^3#real^3)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN  MRESA_TAC face_subset_dart_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN ABBREV_TAC`y1=f1_fan x V E (y:real^3#real^3#real^3#real^3)`
THEN MRESA_TAC condition_f1_fan_in_face_set[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(y1:real^3#real^3#real^3#real^3)` ;`(y:real^3#real^3#real^3#real^3)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN MRESA_TAC f_fan_no_fix_point[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th`(x':real^3#real^3#real^3#real^3)` 
THEN MP_TAC(SET_RULE`y IN face (hypermap1_of_fanx (x,V,E)) x'/\ face (hypermap1_of_fanx (x,V,E)) x' SUBSET d1_fan (x,V,E)==> y IN d1_fan (x,V,E)`) THEN RESA_TAC
THEN MRESA1_TAC th`(y:real^3#real^3#real^3#real^3)`)
THEN DISJ_CASES_TAC(SET_RULE`y1=x' \/ ~(y1=x':real^3#real^3#real^3#real^3)`)
THENL(*2*)[
MP_TAC(SET_RULE`x'=y1:real^3#real^3#real^3#real^3==> pr2 x'= pr2 y1`) 
THEN POP_ASSUM(fun th-> GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN REWRITE_TAC[]
THEN STRIP_TAC
THEN MRESA_TAC properties_of_f1_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(y1:real^3#real^3#real^3#real^3)` ;`(y:real^3#real^3#real^3#real^3)`]
THEN REMOVE_ASSUM_TAC
THEN MRESA_TAC properties_of_f1_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(y:real^3#real^3#real^3#real^3)` ;`(x':real^3#real^3#real^3#real^3)`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC THEN RESA_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` pr2 (y1:real^3#real^3#real^3#real^3)`;
` pr2 (y:real^3#real^3#real^3#real^3)`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN RESA_TAC THEN RESA_TAC
THEN DISJ_CASES_TAC(SET_RULE`set_of_edge (pr2 (y:real^3#real^3#real^3#real^3)) V E = {pr2 y1} \/ ~(set_of_edge (pr2 (y:real^3#real^3#real^3#real^3)) V E = {pr2 y1})`)
THENL(*3*)[
MRESA_TAC CARD_SING[`(pr2 y1):real^3`; `(set_of_edge (pr2 (y:real^3#real^3#real^3#real^3)) V E):real^3->bool`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th->REWRITE_TAC[SYM(th)])
THEN FIND_ASSUM (fun th -> MP_TAC(ISPEC `(pr2 (y:real^3#real^3#real^3#real^3))` th))`(!v:real^3. v IN V==>CARD (set_of_edge v V E) >1)`
THEN FIND_ASSUM (fun th-> REWRITE_TAC[th])`(pr2 (y:real^3#real^3#real^3#real^3)) IN V`
THEN ARITH_TAC;(*3*)
MRESA_TAC SIGMA_FAN [`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` pr2 (y:real^3#real^3#real^3#real^3)`;
` pr2 (y1:real^3#real^3#real^3#real^3)`]
THEN ASM_TAC THEN SET_TAC[]](*2*);
MP_TAC(SET_RULE`x' IN ds /\ y IN ds /\ y1 IN ds==> {x',y,y1:real^3#real^3#real^3#real^3} SUBSET ds`)
THEN RESA_TAC
THEN SUBGOAL_THEN(`CARD {x',y,y1:real^3#real^3#real^3#real^3}=3`)ASSUME_TAC
THENL(*3*)[SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY;
                 IN_INSERT; NOT_IN_EMPTY]

THEN ASM_REWRITE_TAC[]
THEN ARITH_TAC;

  MRESA_TAC FINITE_FACE_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN MRESA_TAC CARD_SUBSET[`{x',y,y1:real^3#real^3#real^3#real^3}`; `ds:real^3#real^3#real^3#real^3->bool`]
]]]);;








let CARD_FACE_SET_EQ_3_FULLY_SURROUNDED_FAN=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ CARD ds =3
==> ?f1 f2 f3. ds={f1,f2,f3}/\ f2=f1_fan x V E f1 /\ f3=f1_fan x V E f2 `,


REPEAT STRIP_TAC
THEN MRESA_TAC hypermap_of_fan_rep[`x:real^3`;`V:real^3->bool`;`(E:(real^3->bool)->bool)`;`(\t. res (t x V E) (d1_fan (x,V,E)))`]
THEN MRESA_TAC dartset_fully_surrounded_is_non_isolated_fan[`x:real^3`;`(V:real^3->bool)`; `(E:(real^3->bool)->bool)`]
THEN MRESA_TAC lemma_face_representation[`hypermap1_of_fanx(x:real^3,V,E)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN SUBGOAL_THEN`x' IN ds:real^3#real^3#real^3#real^3->bool` ASSUME_TAC

THENL (*1*)[
ASM_REWRITE_TAC[face;orbit_map;IN_ELIM_THM;]
THEN EXISTS_TAC`0`
THEN REWRITE_TAC[POWER;I_DEF]
THEN ARITH_TAC;(*1*)
ABBREV_TAC`y=f1_fan x V E (x':real^3#real^3#real^3#real^3)`
THEN MRESA_TAC condition_f1_fan_in_face_set[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(y:real^3#real^3#real^3#real^3)` ;`(x':real^3#real^3#real^3#real^3)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN  MRESA_TAC face_subset_dart_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN ABBREV_TAC`y1=f1_fan x V E (y:real^3#real^3#real^3#real^3)`
THEN MRESA_TAC condition_f1_fan_in_face_set[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(y1:real^3#real^3#real^3#real^3)` ;`(y:real^3#real^3#real^3#real^3)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN MRESA_TAC f_fan_no_fix_point[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th`(x':real^3#real^3#real^3#real^3)` 
THEN MP_TAC(SET_RULE`y IN face (hypermap1_of_fanx (x,V,E)) x'/\ face (hypermap1_of_fanx (x,V,E)) x' SUBSET d1_fan (x,V,E)==> y IN d1_fan (x,V,E)`) THEN RESA_TAC
THEN MRESA1_TAC th`(y:real^3#real^3#real^3#real^3)`)
THEN DISJ_CASES_TAC(SET_RULE`y1=x' \/ ~(y1=x':real^3#real^3#real^3#real^3)`)
THENL(*2*)[
MP_TAC(SET_RULE`x'=y1:real^3#real^3#real^3#real^3==> pr2 x'= pr2 y1`) 
THEN POP_ASSUM(fun th-> GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN REWRITE_TAC[]
THEN STRIP_TAC
THEN MRESA_TAC properties_of_f1_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(y1:real^3#real^3#real^3#real^3)` ;`(y:real^3#real^3#real^3#real^3)`]
THEN REMOVE_ASSUM_TAC
THEN MRESA_TAC properties_of_f1_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(y:real^3#real^3#real^3#real^3)` ;`(x':real^3#real^3#real^3#real^3)`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC THEN RESA_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` pr2 (y1:real^3#real^3#real^3#real^3)`;
` pr2 (y:real^3#real^3#real^3#real^3)`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN RESA_TAC THEN RESA_TAC
THEN DISJ_CASES_TAC(SET_RULE`set_of_edge (pr2 (y:real^3#real^3#real^3#real^3)) V E = {pr2 y1} \/ ~(set_of_edge (pr2 (y:real^3#real^3#real^3#real^3)) V E = {pr2 y1})`)
THENL(*3*)[
MRESA_TAC CARD_SING[`(pr2 y1):real^3`; `(set_of_edge (pr2 (y:real^3#real^3#real^3#real^3)) V E):real^3->bool`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th->REWRITE_TAC[SYM(th)])
THEN FIND_ASSUM (fun th -> MP_TAC(ISPEC `(pr2 (y:real^3#real^3#real^3#real^3))` th))`(!v:real^3. v IN V==>CARD (set_of_edge v V E) >1)`
THEN FIND_ASSUM (fun th-> REWRITE_TAC[th])`(pr2 (y:real^3#real^3#real^3#real^3)) IN V`
THEN ARITH_TAC;(*3*)
MRESA_TAC SIGMA_FAN [`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` pr2 (y:real^3#real^3#real^3#real^3)`;
` pr2 (y1:real^3#real^3#real^3#real^3)`]
THEN ASM_TAC THEN SET_TAC[]](*2*);
MP_TAC(SET_RULE`x' IN ds /\ y IN ds /\ y1 IN ds==> {x',y,y1:real^3#real^3#real^3#real^3} SUBSET ds`)
THEN RESA_TAC
THEN SUBGOAL_THEN(`CARD {x',y,y1:real^3#real^3#real^3#real^3}=3`)ASSUME_TAC
THENL(*3*)[SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY;
                 IN_INSERT; NOT_IN_EMPTY]

THEN ASM_REWRITE_TAC[]
THEN ARITH_TAC;

  MRESA_TAC FINITE_FACE_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN MRESA_TAC CARD_SUBSET_EQ[`{x',y,y1:real^3#real^3#real^3#real^3}`; `ds:real^3#real^3#real^3#real^3->bool`]
THEN EXISTS_TAC`x':real^3#real^3#real^3#real^3`
THEN EXISTS_TAC`y:real^3#real^3#real^3#real^3`
THEN EXISTS_TAC`y1:real^3#real^3#real^3#real^3`
THEN ASM_REWRITE_TAC[]
]]]);;




let lemma_CARD_FACE_SET_EQ_3_FULLY_SURROUNDED_FAN=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds f1 f2 f3.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ CARD ds =3
/\ f1_fan x V E f1=f2 /\ f1_fan x V E f2 =f3 /\ ds={f1,f2,f3}
==> f1=f1_fan x V E f3`,
REPEAT STRIP_TAC
THEN MRESA_TAC dartset_fully_surrounded_is_non_isolated_fan[`x:real^3`;`(V:real^3->bool)`; `(E:(real^3->bool)->bool)`]
THEN  MRESA_TAC face_subset_dart_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN MP_TAC(SET_RULE`{f1, f2, f3} SUBSET d1_fan (x,V,E)==> f1 IN d1_fan (x,V,E) /\ f2 IN d1_fan (x,V,E) /\ f3 IN d1_fan (x,V,E)`) THEN RESA_TAC
THEN ABBREV_TAC`y1=f1_fan x V E (f3:real^3#real^3#real^3#real^3)`
THEN DISJ_CASES_TAC(SET_RULE`y1=f2 \/ ~(y1=f2:real^3#real^3#real^3#real^3)`)
THENL[
MP_TAC(SET_RULE`f2=y1:real^3#real^3#real^3#real^3==> pr2 f2= pr2 y1`) 
THEN POP_ASSUM(fun th-> GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN REWRITE_TAC[]
THEN STRIP_TAC
THEN MRESA_TAC properties_of_f1_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(y1:real^3#real^3#real^3#real^3)` ;`(f3:real^3#real^3#real^3#real^3)`]
THEN REMOVE_ASSUM_TAC
THEN MRESA_TAC properties_of_f1_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(f3:real^3#real^3#real^3#real^3)` ;`(f2:real^3#real^3#real^3#real^3)`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC THEN RESA_TAC
THEN MRESA_TAC remark1_fan[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool) `;` pr2 (y1:real^3#real^3#real^3#real^3)`;
` pr2 (f3:real^3#real^3#real^3#real^3)`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN RESA_TAC THEN RESA_TAC
THEN DISJ_CASES_TAC(SET_RULE`set_of_edge (pr2 (f3:real^3#real^3#real^3#real^3)) V E = {pr2 y1} \/ ~(set_of_edge (pr2 (f3:real^3#real^3#real^3#real^3)) V E = {pr2 y1})`)
THENL[
MRESA_TAC CARD_SING[`(pr2 y1):real^3`; `(set_of_edge (pr2 (f3:real^3#real^3#real^3#real^3)) V E):real^3->bool`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th->REWRITE_TAC[SYM(th)])
THEN FIND_ASSUM (fun th -> MP_TAC(ISPEC `(pr2 (f3:real^3#real^3#real^3#real^3))` th))`(!v:real^3. v IN V==>CARD (set_of_edge v V E) >1)`
THEN FIND_ASSUM (fun th-> REWRITE_TAC[th])`(pr2 (f3:real^3#real^3#real^3#real^3)) IN V`
THEN ARITH_TAC;

MRESA_TAC SIGMA_FAN [`(x:real^3)`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;` pr2 (f3:real^3#real^3#real^3#real^3)`;
` pr2 (y1:real^3#real^3#real^3#real^3)`]
THEN ASM_TAC THEN SET_TAC[]];
MRESA_TAC f_fan_no_fix_point[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th`(f3:real^3#real^3#real^3#real^3)`) 
THEN MRESAL_TAC condition_f1_fan_in_face_set[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(y1:real^3#real^3#real^3#real^3)` ;`(f3:real^3#real^3#real^3#real^3)`;`ds:real^3#real^3#real^3#real^3->bool`][SET_RULE`A IN {B,C,A}`]
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN SET_TAC[]]);;



let CARD_FACE_SET_EQ_3_FULLY_SURROUNDED_FAN1=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ CARD ds =3
==> ?f1 f2 f3. ds={f1,f2,f3}/\ f1_fan x V E f1=f2 /\ f1_fan x V E f2 =f3 /\ f1_fan x V E f3 =f1
/\ {pr2 f2, pr2 f3} IN E
/\ {pr2 f3, pr2 f1 } IN E
/\ {pr2 f1, pr2 f2 } IN E
/\ sigma_fan x V E (pr2 f2) (pr2 f3)=pr2 f1
/\ pr2 f3= pr3 f2
/\ pr2 f2= pr3 f1
/\ pr2 f1= pr3 f3`,

REPEAT STRIP_TAC
THEN MRESA_TAC CARD_FACE_SET_EQ_3_FULLY_SURROUNDED_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN POP_ASSUM (fun th->MP_TAC(SYM th))
THEN POP_ASSUM (fun th->MP_TAC(SYM th))
THEN DISCH_TAC THEN DISCH_TAC
THEN MRESA_TAC dartset_fully_surrounded_is_non_isolated_fan[`x:real^3`;`(V:real^3->bool)`; `(E:(real^3->bool)->bool)`]
THEN  MRESA_TAC face_subset_dart_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN MP_TAC(SET_RULE`{f1, f2, f3} SUBSET d1_fan (x,V,E)==> f1 IN d1_fan (x,V,E) /\ f2 IN d1_fan (x,V,E) /\ f3 IN d1_fan (x,V,E)`) THEN RESA_TAC
THEN MRESA_TAC lemma_CARD_FACE_SET_EQ_3_FULLY_SURROUNDED_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`;`f1:real^3#real^3#real^3#real^3`;`f2:real^3#real^3#real^3#real^3`;`f3:real^3#real^3#real^3#real^3`]
THEN POP_ASSUM (fun th->MP_TAC(SYM th))
THEN DISCH_TAC 
THEN MRESA_TAC properties_of_f1_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(f2:real^3#real^3#real^3#real^3)` ;`(f1:real^3#real^3#real^3#real^3)`]
THEN MRESA_TAC properties_of_f1_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(f3:real^3#real^3#real^3#real^3)` ;`(f2:real^3#real^3#real^3#real^3)`]
THEN EXISTS_TAC`f1:real^3#real^3#real^3#real^3`
THEN EXISTS_TAC`f2:real^3#real^3#real^3#real^3`
THEN EXISTS_TAC`f3:real^3#real^3#real^3#real^3`
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN POP_ASSUM MP_TAC 
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN POP_ASSUM MP_TAC 
THEN POP_ASSUM MP_TAC 
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC properties_of_f1_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(f1:real^3#real^3#real^3#real^3)` ;`(f3:real^3#real^3#real^3#real^3)`]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN ASM_REWRITE_TAC[]);;



let KVQWYDL_lemma10=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ CARD ds=3
==>  aff_gt {x} ({pr2(y)| y IN ds}) = dartset_leads_into_fan x V E ds`,

REPEAT STRIP_TAC
THEN MRESA_TAC CARD_FACE_SET_EQ_3_FULLY_SURROUNDED_FAN1[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN REWRITE_TAC[SET_RULE`{pr2 y | y IN {f1, f2, f3}}= {pr2 f1, pr2 f2, pr2 f3}`]
THEN MRESA_TAC KVQWYDL_lemma1[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(pr2 f1):real^3`; `(pr2 f2):real^3`;`pr2 f3:real^3`]
THEN MRESAL_TAC UNIQUE_DARTSET_LEADS_INTO1_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`;`aff_gt {x} {pr2 f1, pr2 f2, pr2 f3}:real^3->bool`;`f2:real^3#real^3#real^3#real^3`][SET_RULE`A IN {B,A,C}`]);;


let IN_D1_FAN_IMP_EDGE_FAN=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) y.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ y IN d_fan(x,V,E) 
==> {pr2 y, pr3 y} IN E`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC dartset_fully_surrounded_is_non_isolated_fan[`x:real^3`;`(V:real^3->bool)`; `(E:(real^3->bool)->bool)`]
THEN REWRITE_TAC[d1_fan;IN_ELIM_THM]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[pr2;pr3]);;

let EQ_PAIR_IMP_EQ_4_FAN=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) y y1.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ y IN d_fan(x,V,E) 
/\ y1 IN d_fan(x,V,E) 
/\ (pr2 y,pr3 y)= (pr2 y1,pr3 y1)
==> y=y1`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC dartset_fully_surrounded_is_non_isolated_fan[`x:real^3`;`(V:real^3->bool)`; `(E:(real^3->bool)->bool)`]
THEN REWRITE_TAC[PAIR_EQ;IN_ELIM_THM;d1_fan]
THEN STRIP_TAC
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[pr2; pr3]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[]);;
let KVQWYDL_lemma30=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ CARD ds=3

==>  (!y. y IN d_fan(x,V,E) /\ dartset_leads_into_fan x V E ds = dart_leads_into x V E (pr2 y) (pr3 y) 
==> y IN ds)`,


REPEAT STRIP_TAC
THEN MRESA_TAC CARD_FACE_SET_EQ_3_FULLY_SURROUNDED_FAN1[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN MRESA_TAC IN_D1_FAN_IMP_EDGE_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`y:real^3#real^3#real^3#real^3`]
THEN MRESA_TAC KVQWYDL_lemma1[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(pr2 f1):real^3`; `(pr2 f2):real^3`;`pr2 f3:real^3`]
THEN MRESAL_TAC UNIQUE_DARTSET_LEADS_INTO1_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`;`aff_gt {x} {pr2 f1, pr2 f2, pr2 f3}:real^3->bool`;`f2:real^3#real^3#real^3#real^3`][SET_RULE`A IN {B,A,C}`]
THEN MRESA_TAC KVQWYDL_lemma3[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(pr2 f1):real^3`; `(pr2 f2):real^3`;`pr2 f3:real^3`; `(pr2 y):real^3`;`pr3 y:real^3`]
THEN MRESA_TAC face_subset_dart_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN MP_TAC(SET_RULE`{f1, f2, f3} SUBSET d_fan (x,V,E)==> f1 IN d_fan (x,V,E) /\ f2 IN d_fan (x,V,E) /\ f3 IN d_fan (x,V,E)`) THEN RESA_TAC
THEN MP_TAC(SET_RULE`pr2 y,pr3 y IN {(pr2 f1,pr2 f2), (pr2 f2,pr3 f2), (pr3 f2,pr2 f1)}==>
(pr2 y,pr3 y) =(pr2 f1,pr2 f2)\/ (pr2 y,pr3 y) = (pr2 f2,pr3 f2)\/ (pr2 y,pr3 y) =(pr3 f2,pr2 f1)`)
THEN RESA_TAC
THENL[
MRESA_TAC EQ_PAIR_IMP_EQ_4_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`y:real^3#real^3#real^3#real^3`;`f1:real^3#real^3#real^3#real^3`]
THEN SET_TAC[];
MRESA_TAC EQ_PAIR_IMP_EQ_4_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`y:real^3#real^3#real^3#real^3`;`f2:real^3#real^3#real^3#real^3`]
THEN SET_TAC[];
MRESA_TAC EQ_PAIR_IMP_EQ_4_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`y:real^3#real^3#real^3#real^3`;`f3:real^3#real^3#real^3#real^3`]
THEN SET_TAC[]]);;

let KVQWYDL=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ CARD ds=3

==>   aff_gt {x} ({pr2(y)| y IN ds}) = dartset_leads_into_fan x V E ds
/\  (!y. y IN d_fan(x,V,E) /\ dartset_leads_into_fan x V E ds = dart_leads_into x V E (pr2 y) (pr3 y) 
==> y IN ds)`,
MESON_TAC[KVQWYDL_lemma30;KVQWYDL_lemma10]);;


(**************MOZNWEH*******************)


let dartset_leads_into_fan_radial=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds r.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ CARD ds=3
/\ r> &0
==>   radial_norm r x ((dartset_leads_into_fan x V E ds) INTER normball  x r) `,

REPEAT STRIP_TAC
THEN MRESA_TAC KVQWYDL_lemma10[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th)])
THEN MRESA_TAC CARD_FACE_SET_EQ_3_FULLY_SURROUNDED_FAN1[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN REWRITE_TAC[SET_RULE`{pr2 y | y IN {f1, f2, f3}}={pr2 f1,pr2 f2,pr2 f3}`]
THEN MATCH_MP_TAC aff_gt_radial
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (pr2 f2):real^3`;`(pr2 f1):real^3`]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (pr2 f3):real^3`;`(pr2 f2):real^3`]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (pr2 f1):real^3`;`(pr2 f3):real^3`]
THEN ASM_TAC THEN SET_TAC[]);;

let ball_eq_normball=prove(`!x r.ball (x,r)= normball x r`,
REPEAT STRIP_TAC
THEN REWRITE_TAC[ball; normball;DIST_SYM]);;

let dartset_leads_into_fan_eventually_radial_norm=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ CARD ds=3
==>   eventually_radial_norm  x ((dartset_leads_into_fan x V E ds)) `,

REPEAT STRIP_TAC
THEN MRESAL_TAC dartset_leads_into_fan_radial[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`;`&1`][eventually_radial_norm;REAL_ARITH`&1> &0`]
THEN EXISTS_TAC`&1`
THEN ASM_REWRITE_TAC[REAL_ARITH`&1> &0`]);;


let measurable_dartset_leads_into3_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds e:real.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ CARD ds=3
/\ e> &0
==>   measurable((dartset_leads_into_fan x V E ds) INTER ball (x,e))`,
REPEAT STRIP_TAC
THEN MRESA_TAC KVQWYDL_lemma10[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th)])
THEN ONCE_REWRITE_TAC[SET_RULE`A INTER B=B INTER A`]
THEN MESON_TAC[MEASURABLE_BALL_AFF_GT]);;



let CARD_GT1_IMP_AZIM_FAN_EQ_AZIM=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) y.
FAN(x,V,E)
/\ y IN d_fan(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
==>    
azim_fan x V E (pr2 y) (pr3 y)= azim x (pr2 y) (pr3 y) (sigma_fan x V E (pr2 y) (pr3 y))`,
REPEAT STRIP_TAC
THEN REWRITE_TAC[azim_fan]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC "EM")
THEN MRESA_TAC IN_D1_FAN_IMP_EDGE_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`y:real^3#real^3#real^3#real^3`]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (pr3 y):real^3`;`(pr2 y):real^3`]
THEN REMOVE_THEN "EM" (fun th-> MRESA1_TAC th `pr2 (y:real^3#real^3#real^3#real^3)`));;

let CARD_GT1_IMP_AZIM_FAN_EQ_DIHV=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) y.
FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ y IN d_fan(x,V,E)
==>    
azim_fan x V E (pr2 y) (pr3 y)= dihV x (pr2 y) (pr3 y) (sigma_fan x V E (pr2 y) (pr3 y))`,

REPEAT STRIP_TAC
THEN MRESA_TAC CARD_GT1_IMP_AZIM_FAN_EQ_AZIM[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`y:real^3#real^3#real
^3#real^3`]
THEN MRESA_TAC IN_D1_FAN_IMP_EDGE_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`y:real^3#real^3#real^3#real^3`]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` (pr3 y):real^3`;`(pr2 y):real^3`]

THEN MRESA_TAC sigma_fan_in_set_of_edge[`x:real^3`;`V:real^3->bool`;`E:(real^3->bool)->bool`;` (pr2 y):real^3`;`(pr3 y):real^3`]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;`(sigma_fan x V E (pr2 y) (pr3 y)):real^3`;`(pr2 y):real^3`]
THEN POP_ASSUM MP_TAC
THEN RESA_TAC
THEN FIND_ASSUM MP_TAC`fan80(x:real^3,V,E)`
THEN REWRITE_TAC[fan80]
THEN DISCH_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`(pr2 y):real^3`;`(pr3 y):real^3`])
THEN MRESA_TAC AZIM_DIVH[`x:real^3`;`(pr2 y):real^3`;`(pr3 y):real^3`; `(sigma_fan x V E (pr2 y) (pr3 y)):real^3`]);;













let solid_of_dartset_leads_into_fan_triangle_fan=prove( `!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds.
	FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ CARD ds=3
==>    
sol  x (dartset_leads_into_fan x V E ds)= &2 * pi + sum (ds) (\y. (azim_fan x V E (pr2 y) (pr3 y))  - pi)
`,

REPEAT STRIP_TAC
THEN MRESA_TAC CARD_FACE_SET_EQ_3_FULLY_SURROUNDED_FAN1[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"MAS")
THEN REPEAT STRIP_TAC
THEN MRESA_TAC dartset_fully_surrounded_is_non_isolated_fan[`x:real^3`;`(V:real^3->bool)`; `(E:(real^3->bool)->bool)`]
THEN  MRESA_TAC face_subset_dart_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`]
THEN MP_TAC(SET_RULE`{f1, f2, f3} SUBSET d1_fan (x,V,E)==> f1 IN d1_fan (x,V,E) /\ f2 IN d1_fan (x,V,E) /\ f3 IN d1_fan (x,V,E)`) THEN RESA_TAC
THEN MRESA_TAC f_fan_no_fix_point[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th`(f3:real^3#real^3#real^3#real^3)` THEN MRESA1_TAC th`(f2:real^3#real^3#real^3#real^3)` THEN MRESA1_TAC th`(f1:real^3#real^3#real^3#real^3)`) 
THEN MP_TAC(SET_RULE`~(f1 = f3) /\ ~(f3 = f2) /\ ~(f2 = f1)==> DISJOINT {f1:real^3#real^3#real^3#real^3} {f2, f3} /\ DISJOINT {f2:real^3#real^3#real^3#real^3} { f3}`)
THEN RESA_TAC
THEN MRESAL_TAC SUM_UNION[`(\y. azim_fan x V E (pr2 (y:real^3#real^3#real^3#real^3)) (pr3 y) - pi)`;`{(f1:real^3#real^3#real^3#real^3)}`;`{(f2:real^3#real^3#real^3#real^3),(f3:real^3#real^3#real^3#real^3)}`;][SUM_SING]
THEN POP_ASSUM MP_TAC
THEN SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY;
                 IN_INSERT; NOT_IN_EMPTY]

THEN MRESAL_TAC SUM_UNION[`(\y. azim_fan x V E (pr2 (y:real^3#real^3#real^3#real^3)) (pr3 y) - pi)`;`{(f2:real^3#real^3#real^3#real^3)}`;`{(f3:real^3#real^3#real^3#real^3)}`;][SUM_SING]
THEN POP_ASSUM MP_TAC
THEN SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY;
                 IN_INSERT; NOT_IN_EMPTY]
THEN REWRITE_TAC[SET_RULE`{f2} UNION {f3}={f2,f3}`;SET_RULE`{f1} UNION {f2,f3}={f1,f2,f3}`]
THEN RESA_TAC THEN RESA_TAC
THEN MRESAL_TAC measurable_dartset_leads_into3_fan[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`;`&1`][REAL_ARITH`&1> &0`]
THEN MRESAL_TAC dartset_leads_into_fan_radial[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`;`&1`][REAL_ARITH`&1> &0`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ball_eq_normball]
THEN MRESA_TAC KVQWYDL_lemma1[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(pr2 f1):real^3`; `(pr2 f2):real^3`;`pr2 f3:real^3`]

THEN MRESAL_TAC UNIQUE_DARTSET_LEADS_INTO1_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`ds:real^3#real^3#real^3#real^3->bool`;`aff_gt {x} {pr2 f1, pr2 f2, pr2 f3}:real^3->bool`;`f2:real^3#real^3#real^3#real^3`][SET_RULE`A IN {B,A,C}`]
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th;])
THEN MRESA_TAC CARD_GT1_IMP_AZIM_FAN_EQ_DIHV[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`f1:real^3#real^3#real^3#real^3`]
THEN MRESA_TAC CARD_GT1_IMP_AZIM_FAN_EQ_DIHV[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`f2:real^3#real^3#real^3#real^3`]
THEN MRESA_TAC CARD_GT1_IMP_AZIM_FAN_EQ_DIHV[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`f3:real^3#real^3#real^3#real^3`]
THEN REPEAT STRIP_TAC
THEN MRESAL_TAC sol[`x:real^3`;`aff_gt {x} {pr3 f3, pr3 f1, pr3 f2}`;`&1`][REAL_ARITH`&1> &0`;REAL_ARITH`A/ &1 pow 3=A`]
THEN ONCE_REWRITE_TAC[SET_RULE`A INTER B=B INTER A`]
THEN FIND_ASSUM MP_TAC`fan80 (x:real^3,V,E)`
THEN DISCH_TAC
THEN POP_ASSUM (fun th -> MP_TAC th
THEN REWRITE_TAC[fan80] THEN ASSUME_TAC th)
THEN DISCH_TAC
THEN POP_ASSUM (fun th -> MRESA_TAC th [`(pr2 f2):real^3`;`pr2 f3:real^3`]) 
THEN MRESA_TAC properties_fully_surrounded [`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(pr2 f1):real^3`; `(pr2 f2):real^3`;`pr2 f3:real^3`]
THEN MRESAL_TAC VOLUME_SOLID_TRIANGLE[`&1`;`x:real^3`;`(pr2 f1):real^3`; `(pr2 f2):real^3`;`pr2 f3:real^3`][REAL_ARITH`&0 < &1`;ball_eq_normball]
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN MRESA_TAC PROPERTIES_TRIANGLE_FAN[`x:real^3`;`(V:real^3->bool)`;`(E:(real^3->bool)->bool)`;`(pr2 f1):real^3`; `(pr2 f2):real^3`;`pr2 f3:real^3`]
THEN REMOVE_THEN "MAS" MP_TAC
THEN RESA_TAC
THEN REAL_ARITH_TAC);;







let MOZNWEH=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) ds e.
	FAN(x,V,E)
/\ (!v. v IN V==>CARD (set_of_edge v V E) >1)
/\ fan80(x,V,E)
/\ ds IN face_set(hypermap1_of_fanx (x,V,E))
/\ CARD ds=3
/\ e> &0
==>    measurable((dartset_leads_into_fan x V E ds) INTER ball (x,e))
/\ eventually_radial_norm  x ((dartset_leads_into_fan x V E ds))
/\ sol  x (dartset_leads_into_fan x V E ds)= &2 * pi + sum (ds) (\y. (azim_fan x V E (pr2 y) (pr3 y))  - pi)
`,

MESON_TAC[solid_of_dartset_leads_into_fan_triangle_fan;measurable_dartset_leads_into3_fan;dartset_leads_into_fan_eventually_radial_norm]);;
(* open Planarity;; 


*)


end;;