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Theorem btwnconn1lem13 26033
Description: Lemma for btwnconn1 26035. 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 2601 . . 3  |-  ( C  =/=  c  <->  -.  C  =  c )
2 simp2rl 1026 . . . . . . . . . 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 452 . . . . . . . . 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 1024 . . . . . . . . . 10  |-  ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  ->  D  Btwn  <. A ,  c
>. )
54adantr 452 . . . . . . . . 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 519 . . . . . . . 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 960 . . . . . . . . . 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 1002 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
) ) ) )  ->  C  e.  ( EE `  N ) )
9 simpl2 961 . . . . . . . . . 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 741 . . . . . . . . . 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 25948 . . . . . . . . . 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 1186 . . . . . . . . 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 1003 . . . . . . . . . 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 1004 . . . . . . . . . 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 25948 . . . . . . . . . 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 1186 . . . . . . . . 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 692 . . . . . . . 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 211 . . . . . . 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 25880 . . . . . . . 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 1202 . . . . . . 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 42 . . . . . 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 419 . . . . 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 996 . . . . . . . . . 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 452 . . . . . . . . . 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 452 . . . . . . . . . 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 452 . . . . . . . . . 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 25866 . . . . . . . . . 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 1193 . . . . . . . . 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 448 . . . . . . . . . 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 25866 . . . . . . . . . 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 1193 . . . . . . . . 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 2875 . . . . . . . . 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 646 . . . . . . . 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 452 . . . . . . 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 707 . . . . . . . . . . . . 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 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 ) ) )  ->  p  e.  ( EE `  N ) )
37 simprr 734 . . . . . . . . . . . . 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 25866 . . . . . . . . . . . . 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 1193 . . . . . . . . . . . 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 452 . . . . . . . . . . 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 743 . . . . . . . . . . . . . 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 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
) )  ->  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
)  /\  c  e.  ( EE `  N ) ) )
4342ad2antrr 707 . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . . 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 742 . . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . . 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 736 . . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . . 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 519 . . . . . . . . . . . . . 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 737 . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . 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 1020 . . . . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . 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 1021 . . . . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . 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 736 . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . . 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 519 . . . . . . . . . . . . . . 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 997 . . . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . 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 1012 . . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . . 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 1013 . . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . . 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 519 . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . 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 736 . . . . . . . . . . . . . 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 737 . . . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . 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 522 . . . . . . . . . . . . 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 26032 . . . . . . . . . . . . 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 464 . . . . . . . . . . . 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 800 . . . . . . . . . . 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 2834 . . . . . . . . . 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 800 . . . . . . . . 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 589 . . . . . . . 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 2836 . . . . . . 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 800 . . . . 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 2834 . . . 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 599 . . 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 210 . 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 368 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   <.cop 3817   class class class wbr 4212   ` cfv 5454   NNcn 10000   EEcee 25827    Btwn cbtwn 25828  Cgrccgr 25829
This theorem is referenced by:  btwnconn1lem14  26034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-ee 25830  df-btwn 25831  df-cgr 25832  df-ofs 25917  df-ifs 25973  df-cgr3 25974  df-colinear 25975  df-fs 25976
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