
theorem
  for G1 being _Graph, G2 being DLGraphComplement of G1
  for v,w being Vertex of G1 holds
    (not ex e being object st e Joins v,w,G1) implies
    (ex e being object st e Joins v,w,G2)
proof
  let G1 be _Graph, G2 be DLGraphComplement of G1;
  let v,w be Vertex of G1;
  assume A1: not ex e being object st e Joins v,w,G1;
  not ex e being object st e DJoins v,w,G1
  proof
    given e being object such that
      A2: e DJoins v,w,G1;
    e Joins v,w,G1 by A2, GLIB_000:16;
    hence contradiction by A1;
  end;
  then consider e being object such that
    A3: e DJoins v,w,G2 by Def6;
  take e;
  thus e Joins v,w,G2 by A3, GLIB_000:16;
end;
