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Theorem btwnconn1lem13 30878
Description: Lemma for btwnconn1 30880. 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 2626 . . 3  |-  ( C  =/=  c  <->  -.  C  =  c )
2 simp2rl 1078 . . . . . . . . . 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 467 . . . . . . . . 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 1076 . . . . . . . . . 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 467 . . . . . . . . 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 535 . . . . . . . 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 1012 . . . . . . . . . 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 1054 . . . . . . . . . 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 1013 . . . . . . . . . 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 775 . . . . . . . . . 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 30793 . . . . . . . . . 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 1271 . . . . . . . . 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 1055 . . . . . . . . . 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 1056 . . . . . . . . . 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 30793 . . . . . . . . . 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 1271 . . . . . . . . 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 718 . . . . . . . 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 223 . . . . . . 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 24983 . . . . . . . 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 1287 . . . . . . 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 45 . . . . . 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 431 . . . . 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 1048 . . . . . . . . . 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 467 . . . . . . . . . 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 467 . . . . . . . . . 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 467 . . . . . . . . . 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 24969 . . . . . . . . . 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 1278 . . . . . . . . 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 463 . . . . . . . . . 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 24969 . . . . . . . . . 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 1278 . . . . . . . . 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 2960 . . . . . . . . 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 671 . . . . . . . 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 467 . . . . . . 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 733 . . . . . . . . . . . . 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 765 . . . . . . . . . . . . 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 767 . . . . . . . . . . . . 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 24969 . . . . . . . . . . . . 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 1278 . . . . . . . . . . . 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 467 . . . . . . . . . . 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 777 . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . 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 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
) ) )  /\  q  e.  ( EE `  N ) )  -> 
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) ) )
4410ad3antrrr 737 . . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . 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 737 . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . 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 1072 . . . . . . . . . . . . . . . . . . 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 737 . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . 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 1073 . . . . . . . . . . . . . . . . . . 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 737 . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . . . . 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 1061 . . . . . . . . . . . . . . . . 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 737 . . . . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . . 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 1049 . . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . 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 1064 . . . . . . . . . . . . . . . . 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 737 . . . . . . . . . . . . . . . 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 1065 . . . . . . . . . . . . . . . . 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 737 . . . . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . . 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 538 . . . . . . . . . . . . 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 30877 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 836 . . . . . . . . . . 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 2885 . . . . . . . . . 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 836 . . . . . . . . 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 610 . . . . . . . 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 2887 . . . . . . 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 836 . . . . 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 2885 . . . 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 620 . . 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 222 . 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 380 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 188    \/ wo 370    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624   E.wrex 2740   <.cop 3976   class class class wbr 4405   ` cfv 5585   NNcn 10616   EEcee 24930    Btwn cbtwn 24931  Cgrccgr 24932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7961  df-oi 8030  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-sum 13765  df-ee 24933  df-btwn 24934  df-cgr 24935  df-ofs 30762  df-colinear 30818  df-ifs 30819  df-cgr3 30820  df-fs 30821
This theorem is referenced by:  btwnconn1lem14  30879
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