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Theorem btwnconn1lem13 24794
Description: Lemma for btwnconn1 24796. Begin back-filling and eliminating hypotheses. (Contributed by Scott Fenton, 9-Oct-2013.)
Assertion
Ref Expression
btwnconn1lem13  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  ( ( ( A  =/=  B  /\  B  =/=  C
)  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) ) )  ->  ( C  =  c  \/  D  =  d ) )

Proof of Theorem btwnconn1lem13
Dummy variables  e  p  q  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ne 2461 . . 3  |-  ( C  =/=  c  <->  -.  C  =  c )
2 simp2rl 1024 . . . . . . . . . 10  |-  ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  ->  C  Btwn  <. A ,  d
>. )
32adantr 451 . . . . . . . . 9  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  ->  C  Btwn  <. A ,  d
>. )
4 simp2ll 1022 . . . . . . . . . 10  |-  ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  ->  D  Btwn  <. A ,  c
>. )
54adantr 451 . . . . . . . . 9  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  ->  D  Btwn  <. A ,  c
>. )
63, 5jca 518 . . . . . . . 8  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  -> 
( C  Btwn  <. A , 
d >.  /\  D  Btwn  <. A ,  c >. ) )
7 simpl1 958 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  ->  N  e.  NN )
8 simprl1 1000 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  ->  C  e.  ( EE `  N ) )
9 simpl2 959 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  ->  A  e.  ( EE `  N ) )
10 simprrl 740 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  ->  d  e.  ( EE `  N ) )
11 btwncom 24709 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  d  e.  ( EE `  N
) ) )  -> 
( C  Btwn  <. A , 
d >. 
<->  C  Btwn  <. d ,  A >. ) )
127, 8, 9, 10, 11syl13anc 1184 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  ->  ( C  Btwn  <. A ,  d >.  <->  C  Btwn  <. d ,  A >. ) )
13 simprl2 1001 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  ->  D  e.  ( EE `  N ) )
14 simprl3 1002 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  ->  c  e.  ( EE `  N ) )
15 btwncom 24709 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( D  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) ) )  -> 
( D  Btwn  <. A , 
c >. 
<->  D  Btwn  <. c ,  A >. ) )
167, 13, 9, 14, 15syl13anc 1184 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  ->  ( D  Btwn  <. A ,  c >.  <->  D  Btwn  <. c ,  A >. ) )
1712, 16anbi12d 691 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  ->  ( ( C 
Btwn  <. A ,  d
>.  /\  D  Btwn  <. A , 
c >. )  <->  ( C  Btwn  <. d ,  A >.  /\  D  Btwn  <. c ,  A >. ) ) )
186, 17syl5ib 210 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  ->  ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  -> 
( C  Btwn  <. d ,  A >.  /\  D  Btwn  <.
c ,  A >. ) ) )
19 axpasch 24641 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( d  e.  ( EE `  N )  /\  c  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  -> 
( ( C  Btwn  <.
d ,  A >.  /\  D  Btwn  <. c ,  A >. )  ->  E. e  e.  ( EE `  N
) ( e  Btwn  <. C ,  c >.  /\  e  Btwn  <. D , 
d >. ) ) )
207, 10, 14, 9, 8, 13, 19syl132anc 1200 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  ->  ( ( C 
Btwn  <. d ,  A >.  /\  D  Btwn  <. c ,  A >. )  ->  E. e  e.  ( EE `  N
) ( e  Btwn  <. C ,  c >.  /\  e  Btwn  <. D , 
d >. ) ) )
2118, 20syld 40 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  ->  ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  ->  E. e  e.  ( EE `  N ) ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) ) )
2221imp 418 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c ) )  ->  E. e  e.  ( EE `  N ) ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )
23 simpll1 994 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  e  e.  ( EE `  N
) )  ->  N  e.  NN )
2414adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  e  e.  ( EE `  N
) )  ->  c  e.  ( EE `  N
) )
258adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  e  e.  ( EE `  N
) )  ->  C  e.  ( EE `  N
) )
2610adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  e  e.  ( EE `  N
) )  ->  d  e.  ( EE `  N
) )
27 axsegcon 24627 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( c  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  d  e.  ( EE `  N
) ) )  ->  E. p  e.  ( EE `  N ) ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. ) )
2823, 24, 25, 25, 26, 27syl122anc 1191 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  e  e.  ( EE `  N
) )  ->  E. p  e.  ( EE `  N
) ( C  Btwn  <.
c ,  p >.  /\ 
<. C ,  p >.Cgr <. C ,  d >. ) )
29 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  e  e.  ( EE `  N
) )  ->  e  e.  ( EE `  N
) )
30 axsegcon 24627 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( d  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  e  e.  ( EE `  N
) ) )  ->  E. r  e.  ( EE `  N ) ( C  Btwn  <. d ,  r >.  /\  <. C , 
r >.Cgr <. C ,  e
>. ) )
3123, 26, 25, 25, 29, 30syl122anc 1191 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  e  e.  ( EE `  N
) )  ->  E. r  e.  ( EE `  N
) ( C  Btwn  <.
d ,  r >.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) )
32 reeanv 2720 . . . . . . . . . . 11  |-  ( E. p  e.  ( EE
`  N ) E. r  e.  ( EE
`  N ) ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) )  <->  ( E. p  e.  ( EE `  N ) ( C 
Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr
<. C ,  d >.
)  /\  E. r  e.  ( EE `  N
) ( C  Btwn  <.
d ,  r >.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )
3328, 31, 32sylanbrc 645 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  e  e.  ( EE `  N
) )  ->  E. p  e.  ( EE `  N
) E. r  e.  ( EE `  N
) ( ( C 
Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr
<. C ,  d >.
)  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )
3433adantr 451 . . . . . . . . 9  |-  ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  e  e.  ( EE `  N
) )  /\  (
( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A , 
c >.  /\  <. D , 
c >.Cgr <. C ,  D >. )  /\  ( C 
Btwn  <. A ,  d
>.  /\  <. C ,  d
>.Cgr <. C ,  D >. ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) ) )  ->  E. p  e.  ( EE `  N
) E. r  e.  ( EE `  N
) ( ( C 
Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr
<. C ,  d >.
)  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )
357ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  e  e.  ( EE `  N
) )  /\  (
p  e.  ( EE
`  N )  /\  r  e.  ( EE `  N ) ) )  ->  N  e.  NN )
36 simprl 732 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  e  e.  ( EE `  N
) )  /\  (
p  e.  ( EE
`  N )  /\  r  e.  ( EE `  N ) ) )  ->  p  e.  ( EE `  N ) )
37 simprr 733 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  e  e.  ( EE `  N
) )  /\  (
p  e.  ( EE
`  N )  /\  r  e.  ( EE `  N ) ) )  ->  r  e.  ( EE `  N ) )
38 axsegcon 24627 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N ) )  /\  ( r  e.  ( EE `  N )  /\  p  e.  ( EE `  N ) ) )  ->  E. q  e.  ( EE `  N
) ( r  Btwn  <.
p ,  q >.  /\  <. r ,  q
>.Cgr <. r ,  p >. ) )
3935, 36, 37, 37, 36, 38syl122anc 1191 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  e  e.  ( EE `  N
) )  /\  (
p  e.  ( EE
`  N )  /\  r  e.  ( EE `  N ) ) )  ->  E. q  e.  ( EE `  N ) ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )
4039adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  (
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) ) )  ->  E. q  e.  ( EE `  N ) ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )
41 simpl 443 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  ->  ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )
4241ad3antrrr 710 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  (
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  q  e.  ( EE `  N ) )  -> 
( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) ) )
43 simplrl 736 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  e  e.  ( EE `  N
) )  ->  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
)  /\  c  e.  ( EE `  N ) ) )
4443ad2antrr 706 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  (
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  q  e.  ( EE `  N ) )  -> 
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) ) )
4510ad3antrrr 710 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  (
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  q  e.  ( EE `  N ) )  -> 
d  e.  ( EE
`  N ) )
46 simprrr 741 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  ->  b  e.  ( EE `  N ) )
4746ad3antrrr 710 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  (
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  q  e.  ( EE `  N ) )  -> 
b  e.  ( EE
`  N ) )
48 simpllr 735 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  (
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  q  e.  ( EE `  N ) )  -> 
e  e.  ( EE
`  N ) )
4945, 47, 483jca 1132 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  (
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  q  e.  ( EE `  N ) )  -> 
( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )
5044, 49jca 518 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  (
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  q  e.  ( EE `  N ) )  -> 
( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
)  /\  e  e.  ( EE `  N ) ) ) )
51 simplrl 736 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  (
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  q  e.  ( EE `  N ) )  ->  p  e.  ( EE `  N ) )
52 simpr 447 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  (
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  q  e.  ( EE `  N ) )  -> 
q  e.  ( EE
`  N ) )
53 simplrr 737 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  (
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  q  e.  ( EE `  N ) )  -> 
r  e.  ( EE
`  N ) )
5451, 52, 533jca 1132 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  (
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  q  e.  ( EE `  N ) )  -> 
( p  e.  ( EE `  N )  /\  q  e.  ( EE `  N )  /\  r  e.  ( EE `  N ) ) )
5542, 50, 543jca 1132 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  (
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  q  e.  ( EE `  N ) )  -> 
( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N )  /\  e  e.  ( EE `  N
) ) )  /\  ( p  e.  ( EE `  N )  /\  q  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) ) )
56 simp1ll 1018 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  ->  A  =/=  B )
5756ad3antrrr 710 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( ( ( A  =/=  B  /\  B  =/=  C
)  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  ->  A  =/=  B )
5857adantr 451 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )  ->  A  =/=  B )
59 simp1lr 1019 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  ->  B  =/=  C )
6059ad3antrrr 710 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( ( ( A  =/=  B  /\  B  =/=  C
)  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  ->  B  =/=  C )
6160adantr 451 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )  ->  B  =/=  C )
62 simpllr 735 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( ( ( A  =/=  B  /\  B  =/=  C
)  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  ->  C  =/=  c )
6362adantr 451 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )  ->  C  =/=  c )
6458, 61, 633jca 1132 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )  -> 
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c
) )
65 simpl1r 1007 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  -> 
( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )
6665ad3antrrr 710 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )  -> 
( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )
6764, 66jca 518 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )  -> 
( ( A  =/= 
B  /\  B  =/=  C  /\  C  =/=  c
)  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) ) )
68 simpll2 995 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A , 
c >.  /\  <. D , 
c >.Cgr <. C ,  D >. )  /\  ( C 
Btwn  <. A ,  d
>.  /\  <. C ,  d
>.Cgr <. C ,  D >. ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  ->  ( ( D  Btwn  <. A ,  c
>.  /\  <. D ,  c
>.Cgr <. C ,  D >. )  /\  ( C 
Btwn  <. A ,  d
>.  /\  <. C ,  d
>.Cgr <. C ,  D >. ) ) )
6968ad2antrr 706 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )  -> 
( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )
70 simpl3l 1010 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  -> 
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. ) )
7170ad3antrrr 710 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )  -> 
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. ) )
72 simpl3r 1011 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  -> 
( d  Btwn  <. A , 
b >.  /\  <. d ,  b >.Cgr <. D ,  B >. ) )
7372ad3antrrr 710 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )  -> 
( d  Btwn  <. A , 
b >.  /\  <. d ,  b >.Cgr <. D ,  B >. ) )
7471, 73jca 518 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )  -> 
( ( c  Btwn  <. A ,  b >.  /\ 
<. c ,  b >.Cgr <. C ,  B >. )  /\  ( d  Btwn  <. A ,  b >.  /\ 
<. d ,  b >.Cgr <. D ,  B >. ) ) )
7567, 69, 743jca 1132 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )  -> 
( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) ) )
76 simpllr 735 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )  -> 
( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )
77 simplrl 736 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )  -> 
( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. ) )
78 simplrr 737 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )  -> 
( C  Btwn  <. d ,  r >.  /\  <. C ,  r >.Cgr <. C , 
e >. ) )
79 simpr 447 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )  -> 
( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )
8077, 78, 793jca 1132 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )  -> 
( ( C  Btwn  <.
c ,  p >.  /\ 
<. C ,  p >.Cgr <. C ,  d >. )  /\  ( C  Btwn  <.
d ,  r >.  /\  <. C ,  r
>.Cgr <. C ,  e
>. )  /\  (
r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) ) )
8175, 76, 80jca32 521 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )  -> 
( ( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  ( ( e  Btwn  <. C ,  c >.  /\  e  Btwn  <. D , 
d >. )  /\  (
( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. )  /\  (
r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) ) ) ) )
82 btwnconn1lem12 24793 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N )  /\  e  e.  ( EE `  N
) ) )  /\  ( p  e.  ( EE `  N )  /\  q  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  ( ( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  ( ( e  Btwn  <. C ,  c >.  /\  e  Btwn  <. D , 
d >. )  /\  (
( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. )  /\  (
r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) ) ) ) )  ->  D  =  d )
8355, 81, 82syl2an 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  q  e.  ( EE `  N ) )  /\  ( ( ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  /\  ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) ) )  ->  D  =  d )
8483an4s 799 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) ) )  /\  ( q  e.  ( EE `  N
)  /\  ( r  Btwn  <. p ,  q
>.  /\  <. r ,  q
>.Cgr <. r ,  p >. ) ) )  ->  D  =  d )
8584exp32 588 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  (
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) ) )  ->  ( q  e.  ( EE `  N
)  ->  ( (
r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. )  ->  D  =  d ) ) )
8685rexlimdv 2679 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  (
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) ) )  ->  ( E. q  e.  ( EE `  N
) ( r  Btwn  <.
p ,  q >.  /\  <. r ,  q
>.Cgr <. r ,  p >. )  ->  D  =  d ) )
8740, 86mpd 14 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  (
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) ) )  /\  ( ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) ) )  ->  D  =  d )
8887an4s 799 . . . . . . . . . . 11  |-  ( ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  (
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  /\  e  e.  ( EE `  N ) )  /\  ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) ) )  /\  (
( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) ) )  ->  D  =  d )
8988exp32 588 . . . . . . . . . 10  |-  ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  e  e.  ( EE `  N
) )  /\  (
( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A , 
c >.  /\  <. D , 
c >.Cgr <. C ,  D >. )  /\  ( C 
Btwn  <. A ,  d
>.  /\  <. C ,  d
>.Cgr <. C ,  D >. ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) ) )  ->  (
( p  e.  ( EE `  N )  /\  r  e.  ( EE `  N ) )  ->  ( (
( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) )  ->  D  =  d ) ) )
9089rexlimdvv 2686 . . . . . . . . 9  |-  ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  e  e.  ( EE `  N
) )  /\  (
( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A , 
c >.  /\  <. D , 
c >.Cgr <. C ,  D >. )  /\  ( C 
Btwn  <. A ,  d
>.  /\  <. C ,  d
>.Cgr <. C ,  D >. ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) ) )  ->  ( E. p  e.  ( EE `  N ) E. r  e.  ( EE
`  N ) ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) )  ->  D  =  d ) )
9134, 90mpd 14 . . . . . . . 8  |-  ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  e  e.  ( EE `  N
) )  /\  (
( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A , 
c >.  /\  <. D , 
c >.Cgr <. C ,  D >. )  /\  ( C 
Btwn  <. A ,  d
>.  /\  <. C ,  d
>.Cgr <. C ,  D >. ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) ) )  ->  D  =  d )
9291an4s 799 . . . . . . 7  |-  ( ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c ) )  /\  ( e  e.  ( EE `  N
)  /\  ( e  Btwn  <. C ,  c
>.  /\  e  Btwn  <. D , 
d >. ) ) )  ->  D  =  d )
9392exp32 588 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c ) )  ->  ( e  e.  ( EE `  N
)  ->  ( (
e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. )  ->  D  =  d ) ) )
9493rexlimdv 2679 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c ) )  ->  ( E. e  e.  ( EE `  N
) ( e  Btwn  <. C ,  c >.  /\  e  Btwn  <. D , 
d >. )  ->  D  =  d ) )
9522, 94mpd 14 . . . 4  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c ) )  ->  D  =  d )
9695expr 598 . . 3  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  ( ( ( A  =/=  B  /\  B  =/=  C
)  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) ) )  ->  ( C  =/=  c  ->  D  =  d ) )
971, 96syl5bir 209 . 2  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  ( ( ( A  =/=  B  /\  B  =/=  C
)  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. )<