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Theorem btwnconn1lem13 30865
Description: Lemma for btwnconn1 30867. 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 2621 . . 3  |-  ( C  =/=  c  <->  -.  C  =  c )
2 simp2rl 1075 . . . . . . . . . 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 1073 . . . . . . . . . 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 1009 . . . . . . . . . 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 1051 . . . . . . . . . 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 1010 . . . . . . . . . 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 773 . . . . . . . . . 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 30780 . . . . . . . . . 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 1267 . . . . . . . . 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 1052 . . . . . . . . . 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 1053 . . . . . . . . . 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 30780 . . . . . . . . . 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 1267 . . . . . . . . 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 716 . . . . . . . 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 24963 . . . . . . . 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 1283 . . . . . . 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 46 . . . . . 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 1045 . . . . . . . . . 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 24949 . . . . . . . . . 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 1274 . . . . . . . . 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 24949 . . . . . . . . . 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 1274 . . . . . . . . 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 2997 . . . . . . . . 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 669 . . . . . . . 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 731 . . . . . . . . . . . . 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 763 . . . . . . . . . . . . 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 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 ) ) )  ->  r  e.  ( EE `  N ) )
38 axsegcon 24949 . . . . . . . . . . . . 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 1274 . . . . . . . . . . . 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 775 . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . 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 731 . . . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . 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 1069 . . . . . . . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( A  =/=  B  /\  B  =/=  C
)  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  ->  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 1070 . . . . . . . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( A  =/=  B  /\  B  =/=  C
)  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  C  =/=  c )  /\  ( e  Btwn  <. C , 
c >.  /\  e  Btwn  <. D ,  d >. ) )  /\  ( ( C  Btwn  <. c ,  p >.  /\  <. C ,  p >.Cgr <. C ,  d
>. )  /\  ( C  Btwn  <. d ,  r
>.  /\  <. C ,  r
>.Cgr <. C ,  e
>. ) ) )  ->  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 768 . . . . . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . . . . 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 1058 . . . . . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . . . 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 1046 . . . . . . . . . . . . . . . 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 731 . . . . . . . . . . . . . . 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 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 )  -> 
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. ) )
7069ad3antrrr 735 . . . . . . . . . . . . . . . 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 1062 . . . . . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . . 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 30864 . . . . . . . . . . . . 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 834 . . . . . . . . . . 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 2922 . . . . . . . . . 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 834 . . . . . . . . 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 609 . . . . . . . 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 2924 . . . . . . 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 834 . . . . 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 2922 . . . 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 619 . . 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 983    = wceq 1438    e. wcel 1869    =/= wne 2619   E.wrex 2777   <.cop 4003   class class class wbr 4421   ` cfv 5599   NNcn 10611   EEcee 24910    Btwn cbtwn 24911  Cgrccgr 24912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-map 7480  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-sup 7960  df-oi 8029  df-card 8376  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-n0 10872  df-z 10940  df-uz 11162  df-rp 11305  df-ico 11643  df-icc 11644  df-fz 11787  df-fzo 11918  df-seq 12215  df-exp 12274  df-hash 12517  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-clim 13545  df-sum 13746  df-ee 24913  df-btwn 24914  df-cgr 24915  df-ofs 30749  df-colinear 30805  df-ifs 30806  df-cgr3 30807  df-fs 30808
This theorem is referenced by:  btwnconn1lem14  30866
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