Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cgrxfr Structured version   Visualization version   Unicode version
Theorem cgrxfr 32162
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 1064 . . . 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 1117 . . . 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 1116 . . . 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 32134 . . . 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 1326 . . 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 1064 . . . . . . . . 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 477 . . . . . . . . 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 1116 . . . . . . . . 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 1139 . . . . . . . . 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 1140 . . . . . . . . 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 25807 . . . . . . . . 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 1335 . . . . . . . 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 481 . . . . . . 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 681 . . . . . . . . . 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 1064 . . . . . . . . . . . . . 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 794 . . . . . . . . . . . . . 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 796 . . . . . . . . . . . . . 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 1140 . . . . . . . . . . . . . 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 1141 . . . . . . . . . . . . . 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 25807 . . . . . . . . . . . . . 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 1335 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 681 . . . . . . . . . . . . . . . 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 1039 . . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . 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 267 . . . . . . . . . . . . . . 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 801 . . . . . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . . . 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 2849 . . . . . . . . . . . . . . . . . . . . 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 1064 . . . . . . . . . . . . . . . . . . . . . . 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 1067 . . . . . . . . . . . . . . . . . . . . . . 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 1116 . . . . . . . . . . . . . . . . . . . . . . 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 1068 . . . . . . . . . . . . . . . . . . . . . . 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 1069 . . . . . . . . . . . . . . . . . . . . . . 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 794 . . . . . . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . . . . 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 804 . . . . . . . . . . . . . . . . . . . . . . 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 32133 . . . . . . . . . . . . . . . . . . . . . 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 1139 . . . . . . . . . . . . . . . . . . . . . . 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 1140 . . . . . . . . . . . . . . . . . . . . . . 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 1141 . . . . . . . . . . . . . . . . . . . . . . 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 32128 . . . . . . . . . . . . . . . . . . . . . . 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 798 . . . . . . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . . . . 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 796 . . . . . . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . . . . 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 805 . . . . . . . . . . . . . . . . . . . . . . 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 32116 . . . . . . . . . . . . . . . . . . . . . 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 554 . . . . . . . . . . . . . . . . . . . . 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 1117 . . . . . . . . . . . . . . . . . . . . . . 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 800 . . . . . . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . . . . 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 32122 . . . . . . . . . . . . . . . . . . . . . 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 799 . . . . . . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . . . . 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 32098 . . . . . . . . . . . . . . . . . . . . . 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 554 . . . . . . . . . . . . . . . . . . . . 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 1242 . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . . . 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 32117 . . . . . . . . . . . . . . . . . . . 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 1349 . . . . . . . . . . . . . . . . . . 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 47 . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . 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 4403 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( f = F -> <. g , f >. = <. g , F >. )
6564breq2d 4665 . . . . . . . . . . . . . . . . . . . . . 22 |- ( f = F -> ( e Btwn <. g , f >. <-> e Btwn <. g , F >. ) )
66 opeq2 4403 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( f = F -> <. e , f >. = <. e , F >. )
6766breq1d 4663 . . . . . . . . . . . . . . . . . . . . . 22 |- ( f = F -> ( <. e , f >.Cgr <. B , C >. <-> <. e , F >.Cgr <. B , C >. ) )
6865, 67anbi12d 747 . . . . . . . . . . . . . . . . . . . . 21 |- ( f = F -> ( ( e Btwn <. g , f >. /\ <. e , f >.Cgr <. B , C >. ) <-> ( e Btwn <. g , F >. /\ <. e , F >.Cgr <. B , C >. ) ) )
6968biimpa 501 . . . . . . . . . . . . . . . . . . . 20 |- ( ( f = F /\ ( e Btwn <. g , f >. /\ <. e , f >.Cgr <. B , C >. ) ) -> ( e Btwn <. g , F >. /\ <. e , F >.Cgr <. B , C >. ) )
70 simpl 473 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( e Btwn <. g , F >. /\ <. e , F >.Cgr <. B , C >. ) -> e Btwn <. g , F >. )
71 btwnexch3 32127 . . . . . . . . . . . . . . . . . . . . . . . . 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 1335 . . . . . . . . . . . . . . . . . . . . . . . 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 688 . . . . . . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . . . . . . 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 801 . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . 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 32098 . . . . . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . . . . 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 805 . . . . . . . . . . . . . . . . . . . . . . . 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 32098 . . . . . . . . . . . . . . . . . . . . . . 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 32153 . . . . . . . . . . . . . . . . . . . . . . . . 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 1349 . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . 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 1245 . . . . . . . . . . . . . . . . . . . . . 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 554 . . . . . . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . . . . . . 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 34 . . . . . . . . . . . . . . . . . . 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 >. >. ) ) )
8887expcomd 454 . . . . . . . . . . . . . . . . . 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 649 . . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . 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 489 . . . . . . . . . . . . . 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 846 . . . . . . . . . . . . 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 3031 . . . . . . . . . . . 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 643 . . . . . . . . . 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 489 . . . . . . . . 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 846 . . . . . . . 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 3017 . . . . . . 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 643 . . . . 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 846 . . . 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 3031 . . 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 450 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 setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770  Cgr3ccgr3 32143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-ee 25771  df-btwn 25772  df-cgr 25773  df-ofs 32090  df-cgr3 32148
This theorem is referenced by:  btwnxfr  32163  lineext  32183  seglecgr12im  32217  segletr  32221
  Copyright terms: Public domain W3C validator