let G1, G2 be _Graph; :: thesis: for G3 being DSimpleGraph of G1
for G4 being DSimpleGraph of G2
for G5 being DGraphComplement of G1
for G6 being DGraphComplement of G2 st G4 is G3 -Disomorphic holds
G6 is G5 -Disomorphic
let G3 be DSimpleGraph of G1; :: thesis: for G4 being DSimpleGraph of G2
for G5 being DGraphComplement of G1
for G6 being DGraphComplement of G2 st G4 is G3 -Disomorphic holds
G6 is G5 -Disomorphic
let G4 be DSimpleGraph of G2; :: thesis: for G5 being DGraphComplement of G1
for G6 being DGraphComplement of G2 st G4 is G3 -Disomorphic holds
G6 is G5 -Disomorphic
let G5 be DGraphComplement of G1; :: thesis: for G6 being DGraphComplement of G2 st G4 is G3 -Disomorphic holds
G6 is G5 -Disomorphic
let G6 be DGraphComplement of G2; :: thesis: ( G4 is G3 -Disomorphic implies G6 is G5 -Disomorphic )
A1: ( G5 is DGraphComplement of G3 & G6 is DGraphComplement of G4 ) by Th82;
assume G4 is G3 -Disomorphic ; :: thesis: G6 is G5 -Disomorphic
then consider f being directed PVertexMapping of G3,G4 such that
A2: f is Disomorphism by GLIB_011:50;
A3: ( the_Vertices_of G3 = the_Vertices_of G5 & the_Vertices_of G4 = the_Vertices_of G6 ) by A1, Th80;
then reconsider g = f as PartFunc of (the_Vertices_of G5),(the_Vertices_of G6) ;

now :: thesis: for v, w, e being object st v in dom g & w in dom g & e DJoins v,w,G5 holds
ex e6 being object st e6 DJoins g . v,g . w,G6

let v, w, e be object ; :: thesis: ( v in dom g & w in dom g & e DJoins v,w,G5 implies ex e6 being object st e6 DJoins g . v,g . w,G6 )
assume A4: ( v in dom g & w in dom g & e DJoins v,w,G5 ) ; :: thesis: ex e6 being object st e6 DJoins g . v,g . w,G6
then A5: v <> w by GLIB_000:136;
then A6: g . v <> g . w by A2, A4, FUNCT_1:def 4;
A7: for e3 being object holds not e3 DJoins v,w,G3 by A1, A3, A4, A5, Th80;
thus ex e6 being object st e6 DJoins g . v,g . w,G6 :: thesis: verum

proof

assume A8: for e6 being object holds not e6 DJoins g . v,g . w,G6 ; :: thesis: contradiction
( g . v in rng f & g . w in rng f ) by A4, FUNCT_1:3;
then consider e4 being object such that
A9: e4 DJoins g . v,g . w,G4 by A1, A6, A8, Th80;
consider e3 being object such that
A10: e3 DJoins v,w,G3 by A2, A4, A9, GLIB_011:def 4;
thus contradiction by A7, A10; :: thesis: verum

end;

end;

then reconsider g = g as directed PVertexMapping of G5,G6 by GLIB_011:4;

now :: thesis: for v, w, e6 being object st v in dom g & w in dom g & e6 DJoins g . v,g . w,G6 holds
ex e5 being object st e5 DJoins v,w,G5

let v, w, e6 be object ; :: thesis: ( v in dom g & w in dom g & e6 DJoins g . v,g . w,G6 implies ex e5 being object st e5 DJoins v,w,G5 )
assume A11: ( v in dom g & w in dom g & e6 DJoins g . v,g . w,G6 ) ; :: thesis: ex e5 being object st e5 DJoins v,w,G5
A12: g . v <> g . w by A11, GLIB_000:136;
( g . v in rng f & g . w in rng f ) by A11, FUNCT_1:3;
then A13: for e4 being object holds not e4 DJoins g . v,g . w,G4 by A1, A11, A12, Th80;
thus ex e5 being object st e5 DJoins v,w,G5 :: thesis: verum

proof

assume for e5 being object holds not e5 DJoins v,w,G5 ; :: thesis: contradiction
then consider e3 being object such that
A14: e3 DJoins v,w,G3 by A1, A3, A11, A12, Th80;
consider e4 being object such that
A15: e4 DJoins f . v,f . w,G4 by A11, A14, GLIB_011:def 2;
thus contradiction by A13, A15; :: thesis: verum

end;

end;

then g is Dcontinuous by GLIB_011:def 4;
hence G6 is G5 -Disomorphic by A2, A3, GLIB_011:50; :: thesis: verum