let G1 be _Graph; for G2 being LGraphComplement 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 LGraphComplement of G1; 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 ; ( e1 Joins v,w,G1 implies not e2 Joins v,w,G2 )
assume A1:
e1 Joins v,w,G1
; not e2 Joins v,w,G2
then
( v in the_Vertices_of G1 & w in the_Vertices_of G1 )
by GLIB_000:13;
hence
not e2 Joins v,w,G2
by A1, Def7; verum