
theorem
  for G1 being Dsimple _Graph, G2 being DGraphComplement of G1
  holds G1 is DGraphComplement of G2
proof
  let G1 be Dsimple _Graph, G2 be DGraphComplement of G1;
  A1: the_Vertices_of G1 = the_Vertices_of G2 &
    the_Edges_of G1 misses the_Edges_of G2 by Th80;
  now
    let v,w be Vertex of G2;
    assume A2: v <> w;
    v is Vertex of G1 & w is Vertex of G1 by Th80;
    hence (ex e2 being object st e2 DJoins v,w,G2) iff
      (not ex e1 being object st e1 DJoins v,w,G1) by A2, Th80;
  end;
  hence thesis by A1, Th80;
end;
