let G1 be _Graph; :: thesis: for G2 being GraphComplement of G1
for e1, e2, v, w being object st e1 Joins v,w,G1 holds
not e2 Joins v,w,G2
let G2 be GraphComplement of G1; :: thesis: for e1, e2, v, w being object st e1 Joins v,w,G1 holds
not e2 Joins v,w,G2
let e1, e2, v, w be object ; :: thesis: ( e1 Joins v,w,G1 implies not e2 Joins v,w,G2 )
assume A1: e1 Joins v,w,G1 ; :: thesis: not e2 Joins v,w,G2

per cases ( v <> w or v = w ) ;

suppose A2: v <> w ; :: thesis: not e2 Joins v,w,G2

( v in the_Vertices_of G1 & w in the_Vertices_of G1 ) by A1, GLIB_000:13;
hence not e2 Joins v,w,G2 by A1, A2, Th98; :: thesis: verum

end;

suppose v = w ; :: thesis: not e2 Joins v,w,G2

hence not e2 Joins v,w,G2 by GLIB_000:18; :: thesis: verum

end;

end;