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Theorem btwnconn1lem13 29683
Description: Lemma for btwnconn1 29685. 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 2664 . . 3  |-  ( C  =/=  c  <->  -.  C  =  c )
2 simp2rl 1065 . . . . . . . . . 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 465 . . . . . . . . 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 1063 . . . . . . . . . 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 465 . . . . . . . . 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 532 . . . . . . . 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 999 . . . . . . . . . 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 1041 . . . . . . . . . 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 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
) ) ) )  ->  A  e.  ( EE `  N ) )
10 simprrl 763 . . . . . . . . . 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 29598 . . . . . . . . . 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 1230 . . . . . . . . 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 1042 . . . . . . . . . 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 1043 . . . . . . . . . 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 29598 . . . . . . . . . 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 1230 . . . . . . . . 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 710 . . . . . . . 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 219 . . . . . . 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 24076 . . . . . . . 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 1246 . . . . . . 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 44 . . . . . 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 429 . . . . 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 1035 . . . . . . . . . 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
) )  ->  N  e.  NN )
2414adantr 465 . . . . . . . . . 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
) )  ->  c  e.  ( EE `  N
) )
258adantr 465 . . . . . . . . . 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
) )  ->  C  e.  ( EE `  N
) )
2610adantr 465 . . . . . . . . . 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
) )  ->  d  e.  ( EE `  N
) )
27 axsegcon 24062 . . . . . . . . . 10  |-  ( ( 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 1237 . . . . . . . . 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
) )  ->  E. p  e.  ( EE `  N
) ( C  Btwn  <.
c ,  p >.  /\ 
<. C ,  p >.Cgr <. C ,  d >. ) )
29 simpr 461 . . . . . . . . . 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  e.  ( EE `  N
) )
30 axsegcon 24062 . . . . . . . . . 10  |-  ( ( 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 1237 . . . . . . . . 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
) )  ->  E. r  e.  ( EE `  N
) ( C  Btwn  <.
d ,  r >.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) )
32 reeanv 3034 . . . . . . . . 9  |-  ( 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 664 . . . . . . . 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
) )  ->  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 465 . . . . . . 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 ) ) ) )  /\  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 725 . . . . . . . . . . . . 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 ) ) )  ->  N  e.  NN )
36 simprl 755 . . . . . . . . . . . . 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 ) ) )  ->  p  e.  ( EE `  N ) )
37 simprr 756 . . . . . . . . . . . . 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 ) ) )  ->  r  e.  ( EE `  N ) )
38 axsegcon 24062 . . . . . . . . . . . . 13  |-  ( ( 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 1237 . . . . . . . . . . . 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 ) ) )  ->  E. q  e.  ( EE `  N ) ( r  Btwn  <. p ,  q >.  /\  <. r ,  q >.Cgr <. r ,  p >. ) )
4039adantr 465 . . . . . . . . . . 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 ) )  /\  ( 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 simp-4l 765 . . . . . . . . . . . . . 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
) ) )  /\  q  e.  ( EE `  N ) )  -> 
( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) ) )
42 simplrl 759 . . . . . . . . . . . . . . . 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
) )  ->  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
)  /\  c  e.  ( EE `  N ) ) )
4342ad2antrr 725 . . . . . . . . . . . . . . 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
) ) )  /\  q  e.  ( EE `  N ) )  -> 
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) ) )
4410ad3antrrr 729 . . . . . . . . . . . . . . . 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 ) )  -> 
d  e.  ( EE
`  N ) )
45 simprrr 764 . . . . . . . . . . . . . . . . 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
) ) ) )  ->  b  e.  ( EE `  N ) )
4645ad3antrrr 729 . . . . . . . . . . . . . . . 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 ) )  -> 
b  e.  ( EE
`  N ) )
47 simpllr 758 . . . . . . . . . . . . . . . 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 ) )  -> 
e  e.  ( EE
`  N ) )
4844, 46, 473jca 1176 . . . . . . . . . . . . . . 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
) ) )  /\  q  e.  ( EE `  N ) )  -> 
( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )
4943, 48jca 532 . . . . . . . . . . . . . 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
) ) )  /\  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 ) ) ) )
50 simplrl 759 . . . . . . . . . . . . . . 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
) ) )  /\  q  e.  ( EE `  N ) )  ->  p  e.  ( EE `  N ) )
51 simpr 461 . . . . . . . . . . . . . . 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
) ) )  /\  q  e.  ( EE `  N ) )  -> 
q  e.  ( EE
`  N ) )
52 simplrr 760 . . . . . . . . . . . . . . 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
) ) )  /\  q  e.  ( EE `  N ) )  -> 
r  e.  ( EE
`  N ) )
5350, 51, 523jca 1176 . . . . . . . . . . . . . 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
) ) )  /\  q  e.  ( EE `  N ) )  -> 
( p  e.  ( EE `  N )  /\  q  e.  ( EE `  N )  /\  r  e.  ( EE `  N ) ) )
5441, 49, 533jca 1176 . . . . . . . . . . . . 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
) ) )  /\  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
) ) ) )
55 simp1ll 1059 . . . . . . . . . . . . . . . . . . 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 >. ) ) )  ->  A  =/=  B )
5655ad3antrrr 729 . . . . . . . . . . . . . . . . . 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
>. ) ) )  ->  A  =/=  B )
5756adantr 465 . . . . . . . . . . . . . . . . 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 )
58 simp1lr 1060 . . . . . . . . . . . . . . . . . . 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 >. ) ) )  ->  B  =/=  C )
5958ad3antrrr 729 . . . . . . . . . . . . . . . . . 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
>. ) ) )  ->  B  =/=  C )
6059adantr 465 . . . . . . . . . . . . . . . . 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 >. ) )  ->  B  =/=  C )
61 simpllr 758 . . . . . . . . . . . . . . . . . 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
>. ) ) )  ->  C  =/=  c )
6261adantr 465 . . . . . . . . . . . . . . . . 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 >. ) )  ->  C  =/=  c )
6357, 60, 623jca 1176 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( 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
) )
64 simpl1r 1048 . . . . . . . . . . . . . . . . 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 )  -> 
( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )
6564ad3antrrr 729 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( 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 >. ) )
6663, 65jca 532 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( 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 >. ) ) )
67 simpll2 1036 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( 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 >. ) ) )
6867ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( 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 >. ) ) )
69 simpl3l 1051 . . . . . . . . . . . . . . . . 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 )  -> 
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. ) )
7069ad3antrrr 729 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( 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 >. ) )
71 simpl3r 1052 . . . . . . . . . . . . . . . . 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 )  -> 
( d  Btwn  <. A , 
b >.  /\  <. d ,  b >.Cgr <. D ,  B >. ) )
7271ad3antrrr 729 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( 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 >. ) )
7370, 72jca 532 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( 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 >. ) ) )
7466, 68, 733jca 1176 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( 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 >. ) ) ) )
75 simpllr 758 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( 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 >. ) )
76 simplrl 759 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( 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
>. ) )
77 simplrr 760 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( 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 >. ) )
78 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( 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 >. ) )
7976, 77, 783jca 1176 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( 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 >. ) ) )
8074, 75, 79jca32 535 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( 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 >. ) ) ) ) )
81 btwnconn1lem12 29682 . . . . . . . . . . . . 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 )  /\  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 )
8254, 80, 81syl2an 477 . . . . . . . . . . . 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
) ) )  /\  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 )
8382an4s 824 . . . . . . . . . . 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 ) )  /\  ( 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 )
8440, 83rexlimddv 2963 . . . . . . . . . 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 ) )  /\  ( 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 )
8584an4s 824 . . . . . . . . 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 >. ) ) )  /\  (
( 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 )
8685exp32 605 . . . . . . . 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 >. ) ) )  ->  (
( 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 ) ) )
8786rexlimdvv 2965 . . . . . . 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 ) ) ) )  /\  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 ) )
8834, 87mpd 15 . . . . . 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 ) ) ) )  /\  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 )
8988an4s 824 . . . . 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.  ( EE `  N
)  /\  ( e  Btwn  <. C ,  c
>.  /\  e  Btwn  <. D , 
d >. ) ) )  ->  D  =  d )
9022, 89rexlimddv 2963 . . . 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 )
9190expr 615 . . 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 ) )
921, 91syl5bir 218 . 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 >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) ) )  ->  ( -.  C  =  c  ->  D  =  d ) )
9392orrd 378 1  |-  ( ( ( ( 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 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   <.cop 4039   class class class wbr 4453   ` cfv 5594   NNcn 10548   EEcee 24023    Btwn cbtwn 24024  Cgrccgr 24025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-clim 13290  df-sum 13488  df-ee 24026  df-btwn 24027  df-cgr 24028  df-ofs 29567  df-colinear 29623  df-ifs 29624  df-cgr3 29625  df-fs 29626
This theorem is referenced by:  btwnconn1lem14  29684
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