Metamath Proof Explorer

Theorem angmndaddcpbl

Description: Addition of angles is compatible with angle congruence. (Contributed by Thierry Arnoux, 23-Aug-2026)

Ref Expression
Hypotheses angmndadd.p 
|- P = ( Base ` G )
angmndadd.a 
|- A = { d e. ( P ^m ( 0 ..^ 3 ) ) | ( ( d ` 0 ) =/= ( d ` 1 ) /\ ( d ` 1 ) =/= ( d ` 2 ) ) }
angmndadd.i 
|- I = ( Itv ` G )
angmndadd.d 
|- .- = ( dist ` G )
angmndadd.c 
|- .~ = ( cgrA ` G )
angmndadd.l 
|- L = ( LineG ` G )
angmndadd.g 
|- ( ph -> G e. TarskiG )
angmndadd.o 
|- .+ = ( e e. A , f e. A |-> if ( ( e ` 0 ) e. ( ( e ` 1 ) L ( e ` 2 ) ) , <" ( f ` 0 ) ( f ` 1 ) ( iota_ s e. P ( <" ( f ` 2 ) ( f ` 1 ) s "> .~ e /\ ( ( f ` 1 ) .- s ) = ( ( e ` 1 ) .- ( e ` 0 ) ) ) ) "> , <" ( e ` 0 ) ( e ` 1 ) ( iota_ s e. P ( <" ( e ` 2 ) ( e ` 1 ) s "> .~ f /\ ( ( e ` 1 ) .- s ) = ( ( f ` 1 ) .- ( f ` 0 ) ) /\ ( ( ( e ` 1 ) L ( e ` 2 ) ) i^i ( s I ( e ` 0 ) ) ) =/= (/) ) ) "> ) )
angmndaddcpbl.c 
|- ( ph -> C e. A )
angmndaddcpbl.d 
|- ( ph -> D e. A )
angmndaddcpbl.e 
|- ( ph -> E e. A )
angmndaddcpbl.f 
|- ( ph -> F e. A )
angmndaddcpbl.2 
|- ( ph -> E .~ C )
angmndaddcpbl.3 
|- ( ph -> F .~ D )
Assertion angmndaddcpbl 
|- ( ph -> ( E .+ F ) .~ ( C .+ D ) )

Proof

Step Hyp Ref Expression
1 angmndadd.p 
 |-  P = ( Base ` G )
2 angmndadd.a 
 |-  A = { d e. ( P ^m ( 0 ..^ 3 ) ) | ( ( d ` 0 ) =/= ( d ` 1 ) /\ ( d ` 1 ) =/= ( d ` 2 ) ) }
3 angmndadd.i 
 |-  I = ( Itv ` G )
4 angmndadd.d 
 |-  .- = ( dist ` G )
5 angmndadd.c 
 |-  .~ = ( cgrA ` G )
6 angmndadd.l 
 |-  L = ( LineG ` G )
7 angmndadd.g 
 |-  ( ph -> G e. TarskiG )
8 angmndadd.o 
 |-  .+ = ( e e. A , f e. A |-> if ( ( e ` 0 ) e. ( ( e ` 1 ) L ( e ` 2 ) ) , <" ( f ` 0 ) ( f ` 1 ) ( iota_ s e. P ( <" ( f ` 2 ) ( f ` 1 ) s "> .~ e /\ ( ( f ` 1 ) .- s ) = ( ( e ` 1 ) .- ( e ` 0 ) ) ) ) "> , <" ( e ` 0 ) ( e ` 1 ) ( iota_ s e. P ( <" ( e ` 2 ) ( e ` 1 ) s "> .~ f /\ ( ( e ` 1 ) .- s ) = ( ( f ` 1 ) .- ( f ` 0 ) ) /\ ( ( ( e ` 1 ) L ( e ` 2 ) ) i^i ( s I ( e ` 0 ) ) ) =/= (/) ) ) "> ) )
9 angmndaddcpbl.c 
 |-  ( ph -> C e. A )
10 angmndaddcpbl.d 
 |-  ( ph -> D e. A )
11 angmndaddcpbl.e 
 |-  ( ph -> E e. A )
12 angmndaddcpbl.f 
 |-  ( ph -> F e. A )
13 angmndaddcpbl.2 
 |-  ( ph -> E .~ C )
14 angmndaddcpbl.3 
 |-  ( ph -> F .~ D )
15 simp-6r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) -> E = <" x y z "> )
16 15 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> E = <" x y z "> )
17 simp-7r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> F = <" u v w "> )
18 16 17 oveq12d 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> ( E .+ F ) = ( <" x y z "> .+ <" u v w "> ) )
19 7 ad6antr 
 |-  ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) -> G e. TarskiG )
20 19 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) -> G e. TarskiG )
21 20 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> G e. TarskiG )
22 21 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> G e. TarskiG )
23 22 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> G e. TarskiG )
24 simp-7r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> u e. P )
25 24 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> u e. P )
26 simp-6r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> v e. P )
27 26 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> v e. P )
28 simp-5r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> w e. P )
29 28 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> w e. P )
30 simp-6r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> x e. P )
31 30 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> x e. P )
32 31 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> x e. P )
33 simp-5r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> y e. P )
34 33 adantr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) -> y e. P )
35 34 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> y e. P )
36 35 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> y e. P )
37 simp-11r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> z e. P )
38 37 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> z e. P )
39 simpllr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> u =/= v )
40 39 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> u =/= v )
41 simplr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> v =/= w )
42 41 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> v =/= w )
43 simp-9r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> x =/= y )
44 43 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> x =/= y )
45 simp-8r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> y =/= z )
46 45 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> y =/= z )
47 simpr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> x e. ( y L z ) )
48 47 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> x e. ( y L z ) )
49 simpllr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> t e. P )
50 simplr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> <" w v t "> .~ <" x y z "> )
51 simpr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> ( v .- t ) = ( y .- x ) )
52 1 2 3 4 5 6 23 25 27 29 32 36 38 40 42 44 46 8 48 49 50 51 angmndaddov2 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> ( <" x y z "> .+ <" u v w "> ) = <" u v t "> )
53 18 52 eqtrd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> ( E .+ F ) = <" u v t "> )
54 23 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> G e. TarskiG )
55 29 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> w e. P )
56 simp-10r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> k e. P )
57 56 ad10antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> k e. P )
58 57 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> k e. P )
59 simp-6r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) -> i e. P )
60 59 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> i e. P )
61 60 ad10antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> i e. P )
62 61 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> i e. P )
63 simp-11r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> j e. P )
64 63 ad10antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> j e. P )
65 64 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> j e. P )
66 simpllr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> h e. P )
67 25 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> u e. P )
68 27 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> v e. P )
69 49 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> t e. P )
70 42 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> v =/= w )
71 simp-7r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> j =/= k )
72 71 ad10antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> j =/= k )
73 72 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> j =/= k )
74 32 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> x e. P )
75 36 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> y e. P )
76 38 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> z e. P )
77 eqid 
 |-  ( hlG ` G ) = ( hlG ` G )
78 5 a1i 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> .~ = ( cgrA ` G ) )
79 50 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> <" w v t "> .~ <" x y z "> )
80 78 79 breqdi 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> <" w v t "> ( cgrA ` G ) <" x y z "> )
81 1 3 54 77 55 68 69 74 75 76 80 cgracom 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> <" x y z "> ( cgrA ` G ) <" w v t "> )
82 1 3 6 22 31 35 37 43 47 45 lnrot2 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> z e. ( x L y ) )
83 82 orcd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> ( z e. ( x L y ) \/ x = y ) )
84 83 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> ( z e. ( x L y ) \/ x = y ) )
85 1 3 4 54 74 75 76 55 68 69 81 6 84 cgracol 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> ( t e. ( w L v ) \/ w = v ) )
86 70 necomd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> w =/= v )
87 86 neneqd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> -. w = v )
88 85 87 olcnd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> t e. ( w L v ) )
89 1 3 6 54 68 55 69 70 88 lncom 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> t e. ( v L w ) )
90 1 4 3 54 67 69 tgbtwntriv2 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> t e. ( u I t ) )
91 89 90 elind 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> t e. ( ( v L w ) i^i ( u I t ) ) )
92 91 ne0d 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> ( ( v L w ) i^i ( u I t ) ) =/= (/) )
93 simp-11r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) -> a e. P )
94 93 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> a e. P )
95 94 ad10antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> a e. P )
96 95 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> a e. P )
97 simp-10r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) -> b e. P )
98 97 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> b e. P )
99 98 ad10antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> b e. P )
100 99 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> b e. P )
101 simp-9r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) -> c e. P )
102 101 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> c e. P )
103 102 ad10antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> c e. P )
104 103 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> c e. P )
105 simplr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> <" k j h "> .~ <" a b c "> )
106 78 105 breqdi 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> <" k j h "> ( cgrA ` G ) <" a b c "> )
107 1 3 54 77 58 65 66 96 100 104 106 cgracom 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> <" a b c "> ( cgrA ` G ) <" k j h "> )
108 94 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> a e. P )
109 98 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> b e. P )
110 102 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> c e. P )
111 5 a1i 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> .~ = ( cgrA ` G ) )
112 simpllr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> E = <" x y z "> )
113 13 ad6antr 
 |-  ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) -> E .~ C )
114 113 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) -> E .~ C )
115 114 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> E .~ C )
116 simp-9r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) -> C = <" a b c "> )
117 116 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> C = <" a b c "> )
118 115 117 breqtrd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> E .~ <" a b c "> )
119 112 118 eqbrtrrd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> <" x y z "> .~ <" a b c "> )
120 111 119 breqdi 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> <" x y z "> ( cgrA ` G ) <" a b c "> )
121 120 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> <" x y z "> ( cgrA ` G ) <" a b c "> )
122 1 3 4 22 31 35 37 108 109 110 121 6 83 cgracol 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> ( c e. ( a L b ) \/ a = b ) )
123 122 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> ( c e. ( a L b ) \/ a = b ) )
124 1 3 4 54 96 100 104 58 65 66 107 6 123 cgracol 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> ( h e. ( k L j ) \/ k = j ) )
125 73 necomd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> k =/= j )
126 125 neneqd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> -. k = j )
127 124 126 olcnd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> h e. ( k L j ) )
128 1 3 6 54 65 58 66 73 127 lncom 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> h e. ( j L k ) )
129 1 4 3 54 62 66 tgbtwntriv2 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> h e. ( i I h ) )
130 128 129 elind 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> h e. ( ( j L k ) i^i ( i I h ) ) )
131 130 ne0d 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> ( ( j L k ) i^i ( i I h ) ) =/= (/) )
132 17 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> F = <" u v w "> )
133 14 ad6antr 
 |-  ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) -> F .~ D )
134 133 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) -> F .~ D )
135 134 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> F .~ D )
136 135 ad10antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> F .~ D )
137 136 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> F .~ D )
138 132 137 eqbrtrrd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> <" u v w "> .~ D )
139 simp-9r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> D = <" i j k "> )
140 139 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> D = <" i j k "> )
141 140 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> D = <" i j k "> )
142 138 141 breqtrd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> <" u v w "> .~ <" i j k "> )
143 5 eqcomi 
 |-  ( cgrA ` G ) = .~
144 143 a1i 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> ( cgrA ` G ) = .~ )
145 121 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> <" x y z "> ( cgrA ` G ) <" a b c "> )
146 1 3 54 77 74 75 76 96 100 104 145 cgracom 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> <" a b c "> ( cgrA ` G ) <" x y z "> )
147 1 3 54 77 58 65 66 96 100 104 106 74 75 76 146 cgratr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> <" k j h "> ( cgrA ` G ) <" x y z "> )
148 1 3 54 77 58 65 66 74 75 76 147 cgracom 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> <" x y z "> ( cgrA ` G ) <" k j h "> )
149 1 3 54 77 55 68 69 74 75 76 80 58 65 66 148 cgratr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> <" w v t "> ( cgrA ` G ) <" k j h "> )
150 144 149 breqdi 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> <" w v t "> .~ <" k j h "> )
151 1 3 6 5 54 55 58 62 65 66 67 68 69 70 73 92 131 142 150 tgaaddcpbl2 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> <" u v t "> .~ <" i j h "> )
152 117 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> C = <" a b c "> )
153 152 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> C = <" a b c "> )
154 153 141 oveq12d 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> ( C .+ D ) = ( <" a b c "> .+ <" i j k "> ) )
155 simp-8r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> i =/= j )
156 155 ad10antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> i =/= j )
157 156 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> i =/= j )
158 simp-7r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) -> a =/= b )
159 158 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> a =/= b )
160 159 ad10antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> a =/= b )
161 160 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> a =/= b )
162 simp-6r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) -> b =/= c )
163 162 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> b =/= c )
164 163 ad10antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> b =/= c )
165 164 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> b =/= c )
166 163 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> b =/= c )
167 159 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> a =/= b )
168 167 neneqd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> -. a = b )
169 122 168 olcnd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> c e. ( a L b ) )
170 1 3 6 22 109 110 108 166 169 167 lnrot1 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> a e. ( b L c ) )
171 170 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> a e. ( b L c ) )
172 171 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> a e. ( b L c ) )
173 simpr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> ( j .- h ) = ( b .- a ) )
174 1 2 3 4 5 6 54 62 65 58 96 100 104 157 73 161 165 8 172 66 105 173 angmndaddov2 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> ( <" a b c "> .+ <" i j k "> ) = <" i j h "> )
175 154 174 eqtrd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> ( C .+ D ) = <" i j h "> )
176 151 175 breqtrrd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ <" k j h "> .~ <" a b c "> ) /\ ( j .- h ) = ( b .- a ) ) -> <" u v t "> .~ ( C .+ D ) )
177 176 anasss 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) /\ h e. P ) /\ ( <" k j h "> .~ <" a b c "> /\ ( j .- h ) = ( b .- a ) ) ) -> <" u v t "> .~ ( C .+ D ) )
178 1 2 3 4 5 6 23 61 64 57 95 99 103 156 72 160 164 171 angmndaddov2lem 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> E! h e. P ( <" k j h "> .~ <" a b c "> /\ ( j .- h ) = ( b .- a ) ) )
179 reurex 
 |-  ( E! h e. P ( <" k j h "> .~ <" a b c "> /\ ( j .- h ) = ( b .- a ) ) -> E. h e. P ( <" k j h "> .~ <" a b c "> /\ ( j .- h ) = ( b .- a ) ) )
180 178 179 syl 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> E. h e. P ( <" k j h "> .~ <" a b c "> /\ ( j .- h ) = ( b .- a ) ) )
181 177 180 r19.29a 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> <" u v t "> .~ ( C .+ D ) )
182 53 181 eqbrtrd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ <" w v t "> .~ <" x y z "> ) /\ ( v .- t ) = ( y .- x ) ) -> ( E .+ F ) .~ ( C .+ D ) )
183 182 anasss 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) /\ t e. P ) /\ ( <" w v t "> .~ <" x y z "> /\ ( v .- t ) = ( y .- x ) ) ) -> ( E .+ F ) .~ ( C .+ D ) )
184 1 2 3 4 5 6 22 24 26 28 31 35 37 39 41 43 45 47 angmndaddov2lem 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> E! t e. P ( <" w v t "> .~ <" x y z "> /\ ( v .- t ) = ( y .- x ) ) )
185 reurex 
 |-  ( E! t e. P ( <" w v t "> .~ <" x y z "> /\ ( v .- t ) = ( y .- x ) ) -> E. t e. P ( <" w v t "> .~ <" x y z "> /\ ( v .- t ) = ( y .- x ) ) )
186 184 185 syl 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> E. t e. P ( <" w v t "> .~ <" x y z "> /\ ( v .- t ) = ( y .- x ) ) )
187 183 186 r19.29a 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ x e. ( y L z ) ) -> ( E .+ F ) .~ ( C .+ D ) )
188 15 ad8antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> E = <" x y z "> )
189 simp-8r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> F = <" u v w "> )
190 188 189 oveq12d 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> ( E .+ F ) = ( <" x y z "> .+ <" u v w "> ) )
191 21 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> G e. TarskiG )
192 191 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> G e. TarskiG )
193 simp-11r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> u e. P )
194 simp-10r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> v e. P )
195 simp-9r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> w e. P )
196 30 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> x e. P )
197 196 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> x e. P )
198 33 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> y e. P )
199 198 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> y e. P )
200 simp-4r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> z e. P )
201 200 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> z e. P )
202 201 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> z e. P )
203 simp-7r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> u =/= v )
204 simp-6r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> v =/= w )
205 simplr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> x =/= y )
206 205 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> x =/= y )
207 206 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> x =/= y )
208 simp-6r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> y =/= z )
209 208 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> y =/= z )
210 simp-5r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> -. x e. ( y L z ) )
211 simp-4r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> t e. P )
212 simpllr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> <" z y t "> .~ <" u v w "> )
213 simplr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> ( y .- t ) = ( v .- u ) )
214 simpr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> ( ( y L z ) i^i ( t I x ) ) =/= (/) )
215 1 2 3 4 5 6 192 193 194 195 197 199 202 203 204 207 209 8 210 211 212 213 214 angmndaddov1 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> ( <" x y z "> .+ <" u v w "> ) = <" x y t "> )
216 190 215 eqtrd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> ( E .+ F ) = <" x y t "> )
217 192 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> G e. TarskiG )
218 202 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> z e. P )
219 102 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> c e. P )
220 219 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> c e. P )
221 220 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> c e. P )
222 94 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> a e. P )
223 222 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> a e. P )
224 223 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> a e. P )
225 98 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> b e. P )
226 225 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> b e. P )
227 226 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> b e. P )
228 simp-4r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> h e. P )
229 197 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> x e. P )
230 199 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> y e. P )
231 211 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> t e. P )
232 209 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> y =/= z )
233 163 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> b =/= c )
234 233 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> b =/= c )
235 234 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> b =/= c )
236 21 ad8antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) -> G e. TarskiG )
237 236 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) /\ l e. ( t I x ) ) -> G e. TarskiG )
238 simp-5r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) /\ l e. ( t I x ) ) -> t e. P )
239 236 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) -> G e. TarskiG )
240 33 ad8antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) -> y e. P )
241 240 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) -> y e. P )
242 200 ad8antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) -> z e. P )
243 242 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) -> z e. P )
244 simp-9r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) -> y =/= z )
245 244 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) -> y =/= z )
246 1 3 6 239 241 243 245 tgelrnln 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) -> ( y L z ) e. ran L )
247 simpr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) -> l e. ( y L z ) )
248 1 6 3 239 246 247 tglnpt 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) -> l e. P )
249 248 adantr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) /\ l e. ( t I x ) ) -> l e. P )
250 30 ad8antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) -> x e. P )
251 250 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) /\ l e. ( t I x ) ) -> x e. P )
252 simpr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) /\ l e. ( t I x ) ) -> l e. ( t I x ) )
253 1 4 3 237 238 249 251 252 tgbtwncom 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) /\ l e. ( t I x ) ) -> l e. ( x I t ) )
254 236 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) /\ l e. ( x I t ) ) -> G e. TarskiG )
255 250 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) /\ l e. ( x I t ) ) -> x e. P )
256 248 adantr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) /\ l e. ( x I t ) ) -> l e. P )
257 simp-5r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) /\ l e. ( x I t ) ) -> t e. P )
258 simpr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) /\ l e. ( x I t ) ) -> l e. ( x I t ) )
259 1 4 3 254 255 256 257 258 tgbtwncom 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) /\ l e. ( x I t ) ) -> l e. ( t I x ) )
260 253 259 impbida 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ l e. ( y L z ) ) -> ( l e. ( t I x ) <-> l e. ( x I t ) ) )
261 260 pm5.32da 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) -> ( ( l e. ( y L z ) /\ l e. ( t I x ) ) <-> ( l e. ( y L z ) /\ l e. ( x I t ) ) ) )
262 elin 
 |-  ( l e. ( ( y L z ) i^i ( t I x ) ) <-> ( l e. ( y L z ) /\ l e. ( t I x ) ) )
263 elin 
 |-  ( l e. ( ( y L z ) i^i ( x I t ) ) <-> ( l e. ( y L z ) /\ l e. ( x I t ) ) )
264 262 263 bibi12i 
 |-  ( ( l e. ( ( y L z ) i^i ( t I x ) ) <-> l e. ( ( y L z ) i^i ( x I t ) ) ) <-> ( ( l e. ( y L z ) /\ l e. ( t I x ) ) <-> ( l e. ( y L z ) /\ l e. ( x I t ) ) ) )
265 261 264 sylibr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) -> ( l e. ( ( y L z ) i^i ( t I x ) ) <-> l e. ( ( y L z ) i^i ( x I t ) ) ) )
266 265 eqrdv 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) -> ( ( y L z ) i^i ( t I x ) ) = ( ( y L z ) i^i ( x I t ) ) )
267 266 neeq1d 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) -> ( ( ( y L z ) i^i ( t I x ) ) =/= (/) <-> ( ( y L z ) i^i ( x I t ) ) =/= (/) ) )
268 267 biimpa 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> ( ( y L z ) i^i ( x I t ) ) =/= (/) )
269 268 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> ( ( y L z ) i^i ( x I t ) ) =/= (/) )
270 192 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) /\ l e. ( h I a ) ) -> G e. TarskiG )
271 simp-5r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) /\ l e. ( h I a ) ) -> h e. P )
272 192 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) -> G e. TarskiG )
273 226 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) -> b e. P )
274 220 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) -> c e. P )
275 234 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) -> b =/= c )
276 1 3 6 272 273 274 275 tgelrnln 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) -> ( b L c ) e. ran L )
277 simpr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) -> l e. ( b L c ) )
278 1 6 3 272 276 277 tglnpt 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) -> l e. P )
279 278 adantr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) /\ l e. ( h I a ) ) -> l e. P )
280 223 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) /\ l e. ( h I a ) ) -> a e. P )
281 simpr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) /\ l e. ( h I a ) ) -> l e. ( h I a ) )
282 1 4 3 270 271 279 280 281 tgbtwncom 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) /\ l e. ( h I a ) ) -> l e. ( a I h ) )
283 192 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) /\ l e. ( a I h ) ) -> G e. TarskiG )
284 223 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) /\ l e. ( a I h ) ) -> a e. P )
285 278 adantr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) /\ l e. ( a I h ) ) -> l e. P )
286 simp-5r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) /\ l e. ( a I h ) ) -> h e. P )
287 simpr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) /\ l e. ( a I h ) ) -> l e. ( a I h ) )
288 1 4 3 283 284 285 286 287 tgbtwncom 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) /\ l e. ( a I h ) ) -> l e. ( h I a ) )
289 282 288 impbida 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ l e. ( b L c ) ) -> ( l e. ( h I a ) <-> l e. ( a I h ) ) )
290 289 pm5.32da 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) -> ( ( l e. ( b L c ) /\ l e. ( h I a ) ) <-> ( l e. ( b L c ) /\ l e. ( a I h ) ) ) )
291 elin 
 |-  ( l e. ( ( b L c ) i^i ( h I a ) ) <-> ( l e. ( b L c ) /\ l e. ( h I a ) ) )
292 elin 
 |-  ( l e. ( ( b L c ) i^i ( a I h ) ) <-> ( l e. ( b L c ) /\ l e. ( a I h ) ) )
293 291 292 bibi12i 
 |-  ( ( l e. ( ( b L c ) i^i ( h I a ) ) <-> l e. ( ( b L c ) i^i ( a I h ) ) ) <-> ( ( l e. ( b L c ) /\ l e. ( h I a ) ) <-> ( l e. ( b L c ) /\ l e. ( a I h ) ) ) )
294 290 293 sylibr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) -> ( l e. ( ( b L c ) i^i ( h I a ) ) <-> l e. ( ( b L c ) i^i ( a I h ) ) ) )
295 294 eqrdv 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) -> ( ( b L c ) i^i ( h I a ) ) = ( ( b L c ) i^i ( a I h ) ) )
296 295 neeq1d 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) -> ( ( ( b L c ) i^i ( h I a ) ) =/= (/) <-> ( ( b L c ) i^i ( a I h ) ) =/= (/) ) )
297 296 biimpa 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> ( ( b L c ) i^i ( a I h ) ) =/= (/) )
298 188 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> E = <" x y z "> )
299 115 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> E .~ C )
300 299 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> E .~ C )
301 300 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> E .~ C )
302 298 301 eqbrtrrd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> <" x y z "> .~ C )
303 117 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) -> C = <" a b c "> )
304 303 ad8antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> C = <" a b c "> )
305 302 304 breqtrd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> <" x y z "> .~ <" a b c "> )
306 143 a1i 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> ( cgrA ` G ) = .~ )
307 193 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> u e. P )
308 194 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> v e. P )
309 195 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> w e. P )
310 5 a1i 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> .~ = ( cgrA ` G ) )
311 212 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> <" z y t "> .~ <" u v w "> )
312 310 311 breqdi 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> <" z y t "> ( cgrA ` G ) <" u v w "> )
313 60 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> i e. P )
314 313 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> i e. P )
315 314 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> i e. P )
316 63 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> j e. P )
317 316 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> j e. P )
318 317 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> j e. P )
319 56 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> k e. P )
320 319 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> k e. P )
321 320 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> k e. P )
322 simpllr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> <" c b h "> .~ <" i j k "> )
323 310 322 breqdi 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> <" c b h "> ( cgrA ` G ) <" i j k "> )
324 189 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> F = <" u v w "> )
325 135 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> F .~ D )
326 325 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> F .~ D )
327 326 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> F .~ D )
328 324 327 eqbrtrrd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> <" u v w "> .~ D )
329 139 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) -> D = <" i j k "> )
330 329 ad8antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> D = <" i j k "> )
331 328 330 breqtrd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> <" u v w "> .~ <" i j k "> )
332 310 331 breqdi 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> <" u v w "> ( cgrA ` G ) <" i j k "> )
333 1 3 217 77 307 308 309 315 318 321 332 cgracom 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> <" i j k "> ( cgrA ` G ) <" u v w "> )
334 1 3 217 77 221 227 228 315 318 321 323 307 308 309 333 cgratr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> <" c b h "> ( cgrA ` G ) <" u v w "> )
335 1 3 217 77 221 227 228 307 308 309 334 cgracom 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> <" u v w "> ( cgrA ` G ) <" c b h "> )
336 1 3 217 77 218 230 231 307 308 309 312 221 227 228 335 cgratr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> <" z y t "> ( cgrA ` G ) <" c b h "> )
337 306 336 breqdi 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> <" z y t "> .~ <" c b h "> )
338 1 3 6 5 217 218 221 224 227 228 229 230 231 232 235 269 297 305 337 tgaaddcpbl2 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> <" x y t "> .~ <" a b h "> )
339 304 330 oveq12d 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> ( C .+ D ) = ( <" a b c "> .+ <" i j k "> ) )
340 155 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> i =/= j )
341 340 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> i =/= j )
342 341 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> i =/= j )
343 71 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> j =/= k )
344 343 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> j =/= k )
345 344 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> j =/= k )
346 159 ad5antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) -> a =/= b )
347 346 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> a =/= b )
348 347 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> a =/= b )
349 94 ad8antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) -> a e. P )
350 98 ad8antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) -> b e. P )
351 102 ad8antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) -> c e. P )
352 120 ad8antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) -> <" x y z "> ( cgrA ` G ) <" a b c "> )
353 simplr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) -> -. x e. ( y L z ) )
354 244 neneqd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) -> -. y = z )
355 ioran 
 |-  ( -. ( x e. ( y L z ) \/ y = z ) <-> ( -. x e. ( y L z ) /\ -. y = z ) )
356 353 354 355 sylanbrc 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) -> -. ( x e. ( y L z ) \/ y = z ) )
357 1 6 3 236 240 242 250 356 ncolrot2 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) -> -. ( z e. ( x L y ) \/ x = y ) )
358 1 3 4 236 250 240 242 349 350 351 352 6 357 cgrancol 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) -> -. ( c e. ( a L b ) \/ a = b ) )
359 1 6 3 236 349 350 351 358 ncolrot1 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) -> -. ( a e. ( b L c ) \/ b = c ) )
360 359 orsild 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) -> -. a e. ( b L c ) )
361 360 ad3antrrr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> -. a e. ( b L c ) )
362 361 ad4antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> -. a e. ( b L c ) )
363 simplr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> ( b .- h ) = ( j .- i ) )
364 simpr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> ( ( b L c ) i^i ( h I a ) ) =/= (/) )
365 1 2 3 4 5 6 217 315 318 321 224 227 221 342 345 348 235 8 362 228 322 363 364 angmndaddov1 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> ( <" a b c "> .+ <" i j k "> ) = <" a b h "> )
366 339 365 eqtrd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> ( C .+ D ) = <" a b h "> )
367 338 366 breqtrrd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ <" c b h "> .~ <" i j k "> ) /\ ( b .- h ) = ( j .- i ) ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> <" x y t "> .~ ( C .+ D ) )
368 367 3anasss 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) /\ h e. P ) /\ ( <" c b h "> .~ <" i j k "> /\ ( b .- h ) = ( j .- i ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) ) -> <" x y t "> .~ ( C .+ D ) )
369 1 2 3 4 5 6 192 314 317 320 223 226 220 341 344 347 234 361 angmndaddov1lem 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> E! h e. P ( <" c b h "> .~ <" i j k "> /\ ( b .- h ) = ( j .- i ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) )
370 reurex 
 |-  ( E! h e. P ( <" c b h "> .~ <" i j k "> /\ ( b .- h ) = ( j .- i ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) -> E. h e. P ( <" c b h "> .~ <" i j k "> /\ ( b .- h ) = ( j .- i ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) )
371 369 370 syl 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> E. h e. P ( <" c b h "> .~ <" i j k "> /\ ( b .- h ) = ( j .- i ) /\ ( ( b L c ) i^i ( h I a ) ) =/= (/) ) )
372 368 371 r19.29a 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> <" x y t "> .~ ( C .+ D ) )
373 216 372 eqbrtrd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ <" z y t "> .~ <" u v w "> ) /\ ( y .- t ) = ( v .- u ) ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> ( E .+ F ) .~ ( C .+ D ) )
374 373 3anasss 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) /\ t e. P ) /\ ( <" z y t "> .~ <" u v w "> /\ ( y .- t ) = ( v .- u ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) ) -> ( E .+ F ) .~ ( C .+ D ) )
375 21 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) -> G e. TarskiG )
376 simp-7r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) -> u e. P )
377 simp-6r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) -> v e. P )
378 simp-5r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) -> w e. P )
379 30 ad7antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) -> x e. P )
380 34 ad6antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) -> y e. P )
381 simp-11r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) -> z e. P )
382 simpllr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) -> u =/= v )
383 simplr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) -> v =/= w )
384 simp-9r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) -> x =/= y )
385 simp-8r 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) -> y =/= z )
386 simpr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) -> -. x e. ( y L z ) )
387 1 2 3 4 5 6 375 376 377 378 379 380 381 382 383 384 385 386 angmndaddov1lem 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) -> E! t e. P ( <" z y t "> .~ <" u v w "> /\ ( y .- t ) = ( v .- u ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) )
388 reurex 
 |-  ( E! t e. P ( <" z y t "> .~ <" u v w "> /\ ( y .- t ) = ( v .- u ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) -> E. t e. P ( <" z y t "> .~ <" u v w "> /\ ( y .- t ) = ( v .- u ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) )
389 387 388 syl 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) -> E. t e. P ( <" z y t "> .~ <" u v w "> /\ ( y .- t ) = ( v .- u ) /\ ( ( y L z ) i^i ( t I x ) ) =/= (/) ) )
390 374 389 r19.29a 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) /\ -. x e. ( y L z ) ) -> ( E .+ F ) .~ ( C .+ D ) )
391 exmidd 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) -> ( x e. ( y L z ) \/ -. x e. ( y L z ) ) )
392 187 390 391 mpjaodan 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ u =/= v ) /\ v =/= w ) -> ( E .+ F ) .~ ( C .+ D ) )
393 392 anasss 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ F = <" u v w "> ) /\ ( u =/= v /\ v =/= w ) ) -> ( E .+ F ) .~ ( C .+ D ) )
394 393 anasss 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ w e. P ) /\ ( F = <" u v w "> /\ ( u =/= v /\ v =/= w ) ) ) -> ( E .+ F ) .~ ( C .+ D ) )
395 394 r19.29an 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) /\ u e. P ) /\ v e. P ) /\ E. w e. P ( F = <" u v w "> /\ ( u =/= v /\ v =/= w ) ) ) -> ( E .+ F ) .~ ( C .+ D ) )
396 1 fvexi 
 |-  P e. _V
397 396 2 12 elcgrabasi 
 |-  ( ph -> E. u e. P E. v e. P E. w e. P ( F = <" u v w "> /\ ( u =/= v /\ v =/= w ) ) )
398 397 ad9antr 
 |-  ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) -> E. u e. P E. v e. P E. w e. P ( F = <" u v w "> /\ ( u =/= v /\ v =/= w ) ) )
399 398 ad9antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> E. u e. P E. v e. P E. w e. P ( F = <" u v w "> /\ ( u =/= v /\ v =/= w ) ) )
400 395 399 r19.29vva 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ x =/= y ) /\ y =/= z ) -> ( E .+ F ) .~ ( C .+ D ) )
401 400 anasss 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ E = <" x y z "> ) /\ ( x =/= y /\ y =/= z ) ) -> ( E .+ F ) .~ ( C .+ D ) )
402 401 anasss 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ z e. P ) /\ ( E = <" x y z "> /\ ( x =/= y /\ y =/= z ) ) ) -> ( E .+ F ) .~ ( C .+ D ) )
403 402 r19.29an 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) /\ x e. P ) /\ y e. P ) /\ E. z e. P ( E = <" x y z "> /\ ( x =/= y /\ y =/= z ) ) ) -> ( E .+ F ) .~ ( C .+ D ) )
404 396 2 11 elcgrabasi 
 |-  ( ph -> E. x e. P E. y e. P E. z e. P ( E = <" x y z "> /\ ( x =/= y /\ y =/= z ) ) )
405 404 ad3antrrr 
 |-  ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) -> E. x e. P E. y e. P E. z e. P ( E = <" x y z "> /\ ( x =/= y /\ y =/= z ) ) )
406 405 ad9antr 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) -> E. x e. P E. y e. P E. z e. P ( E = <" x y z "> /\ ( x =/= y /\ y =/= z ) ) )
407 403 406 r19.29vva 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ i =/= j ) /\ j =/= k ) -> ( E .+ F ) .~ ( C .+ D ) )
408 407 anasss 
 |-  ( ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ D = <" i j k "> ) /\ ( i =/= j /\ j =/= k ) ) -> ( E .+ F ) .~ ( C .+ D ) )
409 408 anasss 
 |-  ( ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ k e. P ) /\ ( D = <" i j k "> /\ ( i =/= j /\ j =/= k ) ) ) -> ( E .+ F ) .~ ( C .+ D ) )
410 409 r19.29an 
 |-  ( ( ( ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) /\ i e. P ) /\ j e. P ) /\ E. k e. P ( D = <" i j k "> /\ ( i =/= j /\ j =/= k ) ) ) -> ( E .+ F ) .~ ( C .+ D ) )
411 396 2 10 elcgrabasi 
 |-  ( ph -> E. i e. P E. j e. P E. k e. P ( D = <" i j k "> /\ ( i =/= j /\ j =/= k ) ) )
412 411 ad6antr 
 |-  ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) -> E. i e. P E. j e. P E. k e. P ( D = <" i j k "> /\ ( i =/= j /\ j =/= k ) ) )
413 410 412 r19.29vva 
 |-  ( ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ a =/= b ) /\ b =/= c ) -> ( E .+ F ) .~ ( C .+ D ) )
414 413 anasss 
 |-  ( ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ C = <" a b c "> ) /\ ( a =/= b /\ b =/= c ) ) -> ( E .+ F ) .~ ( C .+ D ) )
415 414 anasss 
 |-  ( ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ c e. P ) /\ ( C = <" a b c "> /\ ( a =/= b /\ b =/= c ) ) ) -> ( E .+ F ) .~ ( C .+ D ) )
416 415 r19.29an 
 |-  ( ( ( ( ph /\ a e. P ) /\ b e. P ) /\ E. c e. P ( C = <" a b c "> /\ ( a =/= b /\ b =/= c ) ) ) -> ( E .+ F ) .~ ( C .+ D ) )
417 396 2 9 elcgrabasi 
 |-  ( ph -> E. a e. P E. b e. P E. c e. P ( C = <" a b c "> /\ ( a =/= b /\ b =/= c ) ) )
418 416 417 r19.29vva 
 |-  ( ph -> ( E .+ F ) .~ ( C .+ D ) )