let G1 be non-multi _Graph; :: thesis: for G2 being LGraphComplement of G1 holds G1 is LGraphComplement of G2
let G2 be LGraphComplement of G1; :: thesis: G1 is LGraphComplement of G2
A1: ( the_Vertices_of G1 = the_Vertices_of G2 & the_Edges_of G1 misses the_Edges_of G2 ) by Def7;

now :: thesis: for v, w being Vertex of G2 holds
( ex e2 being object st e2 Joins v,w,G2 iff for e1 being object holds not e1 Joins v,w,G1 )

let v, w be Vertex of G2; :: thesis: ( ex e2 being object st e2 Joins v,w,G2 iff for e1 being object holds not e1 Joins v,w,G1 )
( v is Vertex of G1 & w is Vertex of G1 ) by Def7;
hence ( ex e2 being object st e2 Joins v,w,G2 iff for e1 being object holds not e1 Joins v,w,G1 ) by Def7; :: thesis: verum

end;

hence G1 is LGraphComplement of G2 by A1, Def7; :: thesis: verum