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Theorem cgrxfr 25981
Description: A line segment can be divided at the same place as a congruent line segment is divided. Theorem 4.5 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-Oct-2013.)
Assertion
Ref Expression
cgrxfr  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  -> 
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  ->  E. e  e.  ( EE `  N ) ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )
Distinct variable groups:    A, e    B, e    C, e    D, e   
e, F    e, N

Proof of Theorem cgrxfr
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 960 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. ) )  ->  N  e.  NN )
2 simpl3r 1013 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. ) )  ->  F  e.  ( EE `  N
) )
3 simpl3l 1012 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. ) )  ->  D  e.  ( EE `  N
) )
4 btwndiff 25953 . . . 4  |-  ( ( N  e.  NN  /\  F  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  ->  E. g  e.  ( EE `  N
) ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )
51, 2, 3, 4syl3anc 1184 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. ) )  ->  E. g  e.  ( EE `  N
) ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )
6 simpl1 960 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  ->  N  e.  NN )
7 simpr 448 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  -> 
g  e.  ( EE
`  N ) )
8 simpl3l 1012 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  ->  D  e.  ( EE `  N ) )
9 simpl21 1035 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  ->  A  e.  ( EE `  N ) )
10 simpl22 1036 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  ->  B  e.  ( EE `  N ) )
11 axsegcon 25858 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( g  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  E. e  e.  ( EE `  N ) ( D  Btwn  <. g ,  e >.  /\  <. D , 
e >.Cgr <. A ,  B >. ) )
126, 7, 8, 9, 10, 11syl122anc 1193 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  ->  E. e  e.  ( EE `  N ) ( D  Btwn  <. g ,  e >.  /\  <. D , 
e >.Cgr <. A ,  B >. ) )
1312adantr 452 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  /\  ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) ) )  ->  E. e  e.  ( EE `  N
) ( D  Btwn  <.
g ,  e >.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )
14 anass 631 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  /\  e  e.  ( EE `  N ) )  <->  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) ) )
15 simpl1 960 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  ->  N  e.  NN )
16 simprl 733 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  ->  g  e.  ( EE `  N
) )
17 simprr 734 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  ->  e  e.  ( EE `  N
) )
18 simpl22 1036 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N
) )
19 simpl23 1037 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N
) )
20 axsegcon 25858 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  E. f  e.  ( EE `  N ) ( e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) )
2115, 16, 17, 18, 19, 20syl122anc 1193 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  ->  E. f  e.  ( EE `  N
) ( e  Btwn  <.
g ,  f >.  /\  <. e ,  f
>.Cgr <. B ,  C >. ) )
2221adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  ->  E. f  e.  ( EE `  N ) ( e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) )
23 anass 631 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  /\  f  e.  ( EE `  N
) )  <->  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( ( g  e.  ( EE `  N
)  /\  e  e.  ( EE `  N ) )  /\  f  e.  ( EE `  N
) ) ) )
24 df-3an 938 . . . . . . . . . . . . . . . . 17  |-  ( ( g  e.  ( EE
`  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N
) )  <->  ( (
g  e.  ( EE
`  N )  /\  e  e.  ( EE `  N ) )  /\  f  e.  ( EE `  N ) ) )
2524anbi2i 676 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  <->  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( ( g  e.  ( EE `  N
)  /\  e  e.  ( EE `  N ) )  /\  f  e.  ( EE `  N
) ) ) )
2623, 25bitr4i 244 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  /\  f  e.  ( EE `  N
) )  <->  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) ) )
27 simplrr 738 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  ->  D  =/=  g )
2827ad2antrl 709 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  D  =/=  g
)
2928necomd 2681 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  g  =/=  D
)
30 simpl1 960 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  N  e.  NN )
31 simpr1 963 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  g  e.  ( EE `  N
) )
32 simpl3l 1012 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N
) )
33 simpr2 964 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  e  e.  ( EE `  N
) )
34 simpr3 965 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  f  e.  ( EE `  N
) )
35 simprl 733 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  ->  D  Btwn  <. g ,  e
>. )
3635ad2antrl 709 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  D  Btwn  <. g ,  e >. )
37 simprrl 741 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  e  Btwn  <. g ,  f >. )
3830, 31, 32, 33, 34, 36, 37btwnexchand 25952 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  D  Btwn  <. g ,  f >. )
39 simpl21 1035 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N
) )
40 simpl22 1036 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N
) )
41 simpl23 1037 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N
) )
4230, 31, 32, 33, 34, 36, 37btwnexch3and 25947 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  e  Btwn  <. D , 
f >. )
43 simplll 735 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  ->  B  Btwn  <. A ,  C >. )
4443ad2antrl 709 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  B  Btwn  <. A ,  C >. )
45 simprr 734 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  ->  <. D , 
e >.Cgr <. A ,  B >. )
4645ad2antrl 709 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  <. D ,  e
>.Cgr <. A ,  B >. )
47 simprrr 742 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  <. e ,  f
>.Cgr <. B ,  C >. )
4830, 32, 33, 34, 39, 40, 41, 42, 44, 46, 47cgrextendand 25935 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  <. D ,  f
>.Cgr <. A ,  C >. )
4938, 48jca 519 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  ( D  Btwn  <.
g ,  f >.  /\  <. D ,  f
>.Cgr <. A ,  C >. ) )
50 simpl3r 1013 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  F  e.  ( EE `  N
) )
51 simplrl 737 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  ->  D  Btwn  <. F ,  g
>. )
5251ad2antrl 709 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  D  Btwn  <. F , 
g >. )
5330, 32, 50, 31, 52btwncomand 25941 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  D  Btwn  <. g ,  F >. )
54 simpllr 736 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  ->  <. A ,  C >.Cgr <. D ,  F >. )
5554ad2antrl 709 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  <. A ,  C >.Cgr
<. D ,  F >. )
5630, 39, 41, 32, 50, 55cgrcomand 25917 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  <. D ,  F >.Cgr
<. A ,  C >. )
5753, 56jca 519 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  ( D  Btwn  <.
g ,  F >.  /\ 
<. D ,  F >.Cgr <. A ,  C >. ) )
5829, 49, 573jca 1134 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  ( g  =/= 
D  /\  ( D  Btwn  <. g ,  f
>.  /\  <. D ,  f
>.Cgr <. A ,  C >. )  /\  ( D 
Btwn  <. g ,  F >.  /\  <. D ,  F >.Cgr
<. A ,  C >. ) ) )
5958ex 424 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  (
( ( ( ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. )  /\  ( D 
Btwn  <. F ,  g
>.  /\  D  =/=  g
) )  /\  ( D  Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) )  -> 
( g  =/=  D  /\  ( D  Btwn  <. g ,  f >.  /\  <. D ,  f >.Cgr <. A ,  C >. )  /\  ( D  Btwn  <. g ,  F >.  /\  <. D ,  F >.Cgr
<. A ,  C >. ) ) ) )
60 segconeq 25936 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  NN  /\  ( D  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  (
g  e.  ( EE
`  N )  /\  f  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  -> 
( ( g  =/= 
D  /\  ( D  Btwn  <. g ,  f
>.  /\  <. D ,  f
>.Cgr <. A ,  C >. )  /\  ( D 
Btwn  <. g ,  F >.  /\  <. D ,  F >.Cgr
<. A ,  C >. ) )  ->  f  =  F ) )
6130, 32, 39, 41, 31, 34, 50, 60syl133anc 1207 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  (
( g  =/=  D  /\  ( D  Btwn  <. g ,  f >.  /\  <. D ,  f >.Cgr <. A ,  C >. )  /\  ( D  Btwn  <. g ,  F >.  /\  <. D ,  F >.Cgr
<. A ,  C >. ) )  ->  f  =  F ) )
6259, 61syld 42 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  (
( ( ( ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. )  /\  ( D 
Btwn  <. F ,  g
>.  /\  D  =/=  g
) )  /\  ( D  Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) )  -> 
f  =  F ) )
6362imp 419 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  f  =  F )
64 opeq2 3977 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f  =  F  ->  <. g ,  f >.  =  <. g ,  F >. )
6564breq2d 4216 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  =  F  ->  (
e  Btwn  <. g ,  f >.  <->  e  Btwn  <. g ,  F >. ) )
66 opeq2 3977 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f  =  F  ->  <. e ,  f >.  =  <. e ,  F >. )
6766breq1d 4214 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  =  F  ->  ( <. e ,  f >.Cgr <. B ,  C >.  <->  <. e ,  F >.Cgr <. B ,  C >. ) )
6865, 67anbi12d 692 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f  =  F  ->  (
( e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. )  <->  ( e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr
<. B ,  C >. ) ) )
6968biimpa 471 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  =  F  /\  ( e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) )  -> 
( e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) )
70 simpl 444 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. )  ->  e  Btwn  <.
g ,  F >. )
71 btwnexch3 25946 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  e.  NN  /\  ( g  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( e  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  -> 
( ( D  Btwn  <.
g ,  e >.  /\  e  Btwn  <. g ,  F >. )  ->  e  Btwn  <. D ,  F >. ) )
7230, 31, 32, 33, 50, 71syl122anc 1193 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  (
( D  Btwn  <. g ,  e >.  /\  e  Btwn  <. g ,  F >. )  ->  e  Btwn  <. D ,  F >. ) )
7335, 70, 72syl2ani 638 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  (
( ( ( ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. )  /\  ( D 
Btwn  <. F ,  g
>.  /\  D  =/=  g
) )  /\  ( D  Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) )  ->  e  Btwn  <. D ,  F >. ) )
7473imp 419 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  -> 
e  Btwn  <. D ,  F >. )
75 simplrr 738 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) )  ->  <. D , 
e >.Cgr <. A ,  B >. )
7675adantl 453 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  ->  <. D ,  e >.Cgr <. A ,  B >. )
7730, 32, 33, 39, 40, 76cgrcomand 25917 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  ->  <. A ,  B >.Cgr <. D ,  e >. )
7854ad2antrl 709 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  ->  <. A ,  C >.Cgr <. D ,  F >. )
79 simprrr 742 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  ->  <. e ,  F >.Cgr <. B ,  C >. )
8030, 33, 50, 40, 41, 79cgrcomand 25917 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  ->  <. B ,  C >.Cgr <.
e ,  F >. )
81 brcgr3 25972 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  e  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.  <->  ( <. A ,  B >.Cgr <. D , 
e >.  /\  <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. e ,  F >. ) ) )
8230, 39, 40, 41, 32, 33, 50, 81syl133anc 1207 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.  <->  ( <. A ,  B >.Cgr <. D , 
e >.  /\  <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. e ,  F >. ) ) )
8382adantr 452 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  -> 
( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.  <->  (
<. A ,  B >.Cgr <. D ,  e >.  /\ 
<. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <.
e ,  F >. ) ) )
8477, 78, 80, 83mpbir3and 1137 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
8574, 84jca 519 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  -> 
( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) )
8685expr 599 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  -> 
( ( e  Btwn  <.
g ,  F >.  /\ 
<. e ,  F >.Cgr <. B ,  C >. )  ->  ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
) )
8769, 86syl5 30 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  -> 
( ( f  =  F  /\  ( e 
Btwn  <. g ,  f
>.  /\  <. e ,  f
>.Cgr <. B ,  C >. ) )  ->  (
e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )
8887exp3acom23 1381 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  -> 
( ( e  Btwn  <.
g ,  f >.  /\  <. e ,  f
>.Cgr <. B ,  C >. )  ->  ( f  =  F  ->  ( e 
Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.
) ) ) )
8988impr 603 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  ( f  =  F  ->  ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.
) ) )
9063, 89mpd 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
)
9190expr 599 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  -> 
( ( e  Btwn  <.
g ,  f >.  /\  <. e ,  f
>.Cgr <. B ,  C >. )  ->  ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.
) ) )
9226, 91sylanb 459 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  /\  f  e.  ( EE `  N
) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  -> 
( ( e  Btwn  <.
g ,  f >.  /\  <. e ,  f
>.Cgr <. B ,  C >. )  ->  ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.
) ) )
9392an32s 780 . . . . . . . . . . . . 13  |-  ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  /\  f  e.  ( EE `  N ) )  -> 
( ( e  Btwn  <.
g ,  f >.  /\  <. e ,  f
>.Cgr <. B ,  C >. )  ->  ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.
) ) )
9493rexlimdva 2822 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  -> 
( E. f  e.  ( EE `  N
) ( e  Btwn  <.
g ,  f >.  /\  <. e ,  f
>.Cgr <. B ,  C >. )  ->  ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.
) ) )
9522, 94mpd 15 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  -> 
( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) )
9695expr 599 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. )  /\  ( D 
Btwn  <. F ,  g
>.  /\  D  =/=  g
) ) )  -> 
( ( D  Btwn  <.
g ,  e >.  /\  <. D ,  e
>.Cgr <. A ,  B >. )  ->  ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.
) ) )
9714, 96sylanb 459 . . . . . . . . 9  |-  ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  /\  e  e.  ( EE `  N ) )  /\  ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) ) )  ->  (
( D  Btwn  <. g ,  e >.  /\  <. D ,  e >.Cgr <. A ,  B >. )  ->  (
e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )
9897an32s 780 . . . . . . . 8  |-  ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  /\  ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) ) )  /\  e  e.  ( EE `  N
) )  ->  (
( D  Btwn  <. g ,  e >.  /\  <. D ,  e >.Cgr <. A ,  B >. )  ->  (
e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )
9998reximdva 2810 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  /\  ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) ) )  ->  ( E. e  e.  ( EE `  N ) ( D  Btwn  <. g ,  e >.  /\  <. D , 
e >.Cgr <. A ,  B >. )  ->  E. e  e.  ( EE `  N
) ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
) )
10013, 99mpd 15 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  /\  ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) ) )  ->  E. e  e.  ( EE `  N
) ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
)
101100expr 599 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. ) )  ->  (
( D  Btwn  <. F , 
g >.  /\  D  =/=  g )  ->  E. e  e.  ( EE `  N
) ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
) )
102101an32s 780 . . . 4  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. ) )  /\  g  e.  ( EE `  N
) )  ->  (
( D  Btwn  <. F , 
g >.  /\  D  =/=  g )  ->  E. e  e.  ( EE `  N
) ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
) )
103102rexlimdva 2822 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. ) )  ->  ( E. g  e.  ( EE `  N ) ( D  Btwn  <. F , 
g >.  /\  D  =/=  g )  ->  E. e  e.  ( EE `  N
) ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
) )
1045, 103mpd 15 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. ) )  ->  E. e  e.  ( EE `  N
) ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
)
105104ex 424 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  -> 
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  ->  E. e  e.  ( EE `  N ) ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   <.cop 3809   class class class wbr 4204   ` cfv 5446   NNcn 9992   EEcee 25819    Btwn cbtwn 25820  Cgrccgr 25821  Cgr3ccgr3 25962
This theorem is referenced by:  btwnxfr  25982  lineext  26002  seglecgr12im  26036  segletr  26040
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-ee 25822  df-btwn 25823  df-cgr 25824  df-ofs 25909  df-cgr3 25966
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