Metamath Proof Explorer

Theorem ax5seg

Description: The five segment axiom. Take two triangles A D C and E H G , a point B on A C , and a point F on E G . If all corresponding line segments except for C D and G H are congruent, then so are C D and G H . Axiom A5 of Schwabhauser p. 11. (Contributed by Scott Fenton, 12-Jun-2013)

Ref Expression
Assertion ax5seg 
|- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( ( A =/= B /\ B Btwn <. A , C >. /\ F Btwn <. E , G >. ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) -> <. C , D >. Cgr <. G , H >. ) )

Proof

Step Hyp Ref Expression
1 fzfid 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( 1 ... N ) e. Fin )
2 simpl21 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> C e. ( EE ` N ) )
3 fveere 
 |-  ( ( C e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( C ` j ) e. RR )
4 2 3 sylancom 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( C ` j ) e. RR )
5 simpl22 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> D e. ( EE ` N ) )
6 fveere 
 |-  ( ( D e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( D ` j ) e. RR )
7 5 6 sylancom 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( D ` j ) e. RR )
8 4 7 resubcld 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( ( C ` j ) - ( D ` j ) ) e. RR )
9 8 resqcld 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) e. RR )
10 1 9 fsumrecl 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) e. RR )
11 10 recnd 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) e. CC )
12 11 adantr 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) e. CC )
13 simpl32 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> G e. ( EE ` N ) )
14 fveere 
 |-  ( ( G e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( G ` j ) e. RR )
15 13 14 sylancom 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( G ` j ) e. RR )
16 simpl33 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> H e. ( EE ` N ) )
17 fveere 
 |-  ( ( H e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( H ` j ) e. RR )
18 16 17 sylancom 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( H ` j ) e. RR )
19 15 18 resubcld 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( ( G ` j ) - ( H ` j ) ) e. RR )
20 19 resqcld 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) e. RR )
21 1 20 fsumrecl 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) e. RR )
22 21 recnd 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) e. CC )
23 22 adantr 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) e. CC )
24 elicc01 
 |-  ( t e. ( 0 [,] 1 ) <-> ( t e. RR /\ 0 <_ t /\ t <_ 1 ) )
25 24 simp1bi 
 |-  ( t e. ( 0 [,] 1 ) -> t e. RR )
26 25 recnd 
 |-  ( t e. ( 0 [,] 1 ) -> t e. CC )
27 26 ad2antrr 
 |-  ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) -> t e. CC )
28 27 3ad2ant1 
 |-  ( ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) -> t e. CC )
29 28 adantl 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> t e. CC )
30 simpl11 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> N e. NN )
31 simp12 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
32 simp13 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
33 simp21 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
34 31 32 33 3jca 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) )
35 34 adantr 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) )
36 simprrl 
 |-  ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) -> A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) )
37 36 3ad2ant1 
 |-  ( ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) -> A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) )
38 37 adantl 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) )
39 simp1rl 
 |-  ( ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) -> A =/= B )
40 39 adantl 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> A =/= B )
41 ax5seglem4 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A =/= B ) -> t =/= 0 )
42 30 35 38 40 41 syl211anc 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> t =/= 0 )
43 simpr3r 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> <. B , D >. Cgr <. F , H >. )
44 simpl13 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> B e. ( EE ` N ) )
45 simpl22 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> D e. ( EE ` N ) )
46 simpl31 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> F e. ( EE ` N ) )
47 simpl33 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> H e. ( EE ` N ) )
48 brcgr 
 |-  ( ( ( B e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. B , D >. Cgr <. F , H >. <-> sum_ j e. ( 1 ... N ) ( ( ( B ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( F ` j ) - ( H ` j ) ) ^ 2 ) ) )
49 44 45 46 47 48 syl22anc 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( <. B , D >. Cgr <. F , H >. <-> sum_ j e. ( 1 ... N ) ( ( ( B ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( F ` j ) - ( H ` j ) ) ^ 2 ) ) )
50 43 49 mpbid 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( B ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( F ` j ) - ( H ` j ) ) ^ 2 ) )
51 simp23 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> E e. ( EE ` N ) )
52 simp31 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> F e. ( EE ` N ) )
53 simp32 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> G e. ( EE ` N ) )
54 51 52 53 3jca 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ G e. ( EE ` N ) ) )
55 34 54 jca 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ G e. ( EE ` N ) ) ) )
56 55 adantr 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ G e. ( EE ` N ) ) ) )
57 simpr1l 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) )
58 simprrr 
 |-  ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) -> A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) )
59 58 3ad2ant1 
 |-  ( ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) -> A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) )
60 59 adantl 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) )
61 38 60 jca 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) )
62 simpr2l 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> <. A , B >. Cgr <. E , F >. )
63 simpr2r 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> <. B , C >. Cgr <. F , G >. )
64 ax5seglem6 
 |-  ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ G e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) ) -> t = s )
65 30 56 40 57 61 62 63 64 syl232anc 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> t = s )
66 65 oveq2d 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( 1 - t ) = ( 1 - s ) )
67 54 adantr 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ G e. ( EE ` N ) ) )
68 ax5seglem3 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ G e. ( EE ` N ) ) ) /\ ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( G ` j ) ) ^ 2 ) )
69 30 35 67 57 61 62 63 68 syl322anc 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( G ` j ) ) ^ 2 ) )
70 65 69 oveq12d 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( t x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) = ( s x. sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( G ` j ) ) ^ 2 ) ) )
71 simpr3l 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> <. A , D >. Cgr <. E , H >. )
72 simpl12 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> A e. ( EE ` N ) )
73 simpl23 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> E e. ( EE ` N ) )
74 brcgr 
 |-  ( ( ( A e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. A , D >. Cgr <. E , H >. <-> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( H ` j ) ) ^ 2 ) ) )
75 72 45 73 47 74 syl22anc 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( <. A , D >. Cgr <. E , H >. <-> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( H ` j ) ) ^ 2 ) ) )
76 71 75 mpbid 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( H ` j ) ) ^ 2 ) )
77 70 76 oveq12d 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( ( t x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( D ` j ) ) ^ 2 ) ) = ( ( s x. sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( G ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( H ` j ) ) ^ 2 ) ) )
78 66 77 oveq12d 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( ( 1 - t ) x. ( ( t x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( D ` j ) ) ^ 2 ) ) ) = ( ( 1 - s ) x. ( ( s x. sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( G ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( H ` j ) ) ^ 2 ) ) ) )
79 50 78 oveq12d 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( sum_ j e. ( 1 ... N ) ( ( ( B ` j ) - ( D ` j ) ) ^ 2 ) + ( ( 1 - t ) x. ( ( t x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( D ` j ) ) ^ 2 ) ) ) ) = ( sum_ j e. ( 1 ... N ) ( ( ( F ` j ) - ( H ` j ) ) ^ 2 ) + ( ( 1 - s ) x. ( ( s x. sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( G ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( H ` j ) ) ^ 2 ) ) ) ) )
80 31 32 jca 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) )
81 simp22 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
82 80 33 81 jca32 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) )
83 82 adantr 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) )
84 simp1ll 
 |-  ( ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) -> t e. ( 0 [,] 1 ) )
85 84 adantl 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> t e. ( 0 [,] 1 ) )
86 ax5seglem9 
 |-  ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) ) /\ ( t e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) ) -> ( t x. sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) ) = ( sum_ j e. ( 1 ... N ) ( ( ( B ` j ) - ( D ` j ) ) ^ 2 ) + ( ( 1 - t ) x. ( ( t x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( D ` j ) ) ^ 2 ) ) ) ) )
87 30 83 85 38 86 syl22anc 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( t x. sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) ) = ( sum_ j e. ( 1 ... N ) ( ( ( B ` j ) - ( D ` j ) ) ^ 2 ) + ( ( 1 - t ) x. ( ( t x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( D ` j ) ) ^ 2 ) ) ) ) )
88 3simpc 
 |-  ( ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) -> ( G e. ( EE ` N ) /\ H e. ( EE ` N ) ) )
89 88 3ad2ant3 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( G e. ( EE ` N ) /\ H e. ( EE ` N ) ) )
90 51 52 89 jca31 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( E e. ( EE ` N ) /\ F e. ( EE ` N ) ) /\ ( G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) )
91 90 adantr 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( ( E e. ( EE ` N ) /\ F e. ( EE ` N ) ) /\ ( G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) )
92 simp1lr 
 |-  ( ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) -> s e. ( 0 [,] 1 ) )
93 92 adantl 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> s e. ( 0 [,] 1 ) )
94 ax5seglem9 
 |-  ( ( ( N e. NN /\ ( ( E e. ( EE ` N ) /\ F e. ( EE ` N ) ) /\ ( G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) ) /\ ( s e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) -> ( s x. sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) = ( sum_ j e. ( 1 ... N ) ( ( ( F ` j ) - ( H ` j ) ) ^ 2 ) + ( ( 1 - s ) x. ( ( s x. sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( G ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( H ` j ) ) ^ 2 ) ) ) ) )
95 30 91 93 60 94 syl22anc 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( s x. sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) = ( sum_ j e. ( 1 ... N ) ( ( ( F ` j ) - ( H ` j ) ) ^ 2 ) + ( ( 1 - s ) x. ( ( s x. sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( G ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( H ` j ) ) ^ 2 ) ) ) ) )
96 79 87 95 3eqtr4d 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( t x. sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) ) = ( s x. sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) )
97 65 oveq1d 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( t x. sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) = ( s x. sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) )
98 96 97 eqtr4d 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> ( t x. sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) ) = ( t x. sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) )
99 12 23 29 42 98 mulcanad 
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) )
100 99 3exp2 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) -> ( ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) -> ( ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) -> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) ) ) )
101 100 expd 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> ( ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) -> ( ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) -> ( ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) -> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) ) ) ) )
102 101 rexlimdvv 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) -> ( ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) -> ( ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) -> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) ) ) )
103 102 3impd 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) -> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) )
104 brbtwn 
 |-  ( ( B e. ( EE ` N ) /\ A e. ( EE ` N ) /\ C e. ( EE ` N ) ) -> ( B Btwn <. A , C >. <-> E. t e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) )
105 32 31 33 104 syl3anc 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( B Btwn <. A , C >. <-> E. t e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) )
106 brbtwn 
 |-  ( ( F e. ( EE ` N ) /\ E e. ( EE ` N ) /\ G e. ( EE ` N ) ) -> ( F Btwn <. E , G >. <-> E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) )
107 52 51 53 106 syl3anc 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( F Btwn <. E , G >. <-> E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) )
108 105 107 anbi12d 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) <-> ( E. t e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) )
109 reeanv 
 |-  ( E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) <-> ( E. t e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) )
110 108 109 bitr4di 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) <-> E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) )
111 110 anbi2d 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( A =/= B /\ ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) ) <-> ( A =/= B /\ E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) )
112 3anass 
 |-  ( ( A =/= B /\ B Btwn <. A , C >. /\ F Btwn <. E , G >. ) <-> ( A =/= B /\ ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) ) )
113 r19.42v 
 |-  ( E. s e. ( 0 [,] 1 ) ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) <-> ( A =/= B /\ E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) )
114 113 rexbii 
 |-  ( E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) <-> E. t e. ( 0 [,] 1 ) ( A =/= B /\ E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) )
115 r19.42v 
 |-  ( E. t e. ( 0 [,] 1 ) ( A =/= B /\ E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) <-> ( A =/= B /\ E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) )
116 114 115 bitri 
 |-  ( E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) <-> ( A =/= B /\ E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) )
117 111 112 116 3bitr4g 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( A =/= B /\ B Btwn <. A , C >. /\ F Btwn <. E , G >. ) <-> E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) )
118 117 3anbi1d 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( ( A =/= B /\ B Btwn <. A , C >. /\ F Btwn <. E , G >. ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) <-> ( E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) ) )
119 simp33 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> H e. ( EE ` N ) )
120 brcgr 
 |-  ( ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. C , D >. Cgr <. G , H >. <-> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) )
121 33 81 53 119 120 syl22anc 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. C , D >. Cgr <. G , H >. <-> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) )
122 103 118 121 3imtr4d 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( ( A =/= B /\ B Btwn <. A , C >. /\ F Btwn <. E , G >. ) /\ ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) ) -> <. C , D >. Cgr <. G , H >. ) )