
theorem Th81:
  for G1 being _Graph, G2 being DGraphComplement of G1, e1,e2,v,w being object
  holds e1 DJoins v,w,G1 implies not e2 DJoins v,w,G2
proof
  let G1 be _Graph, G2 be DGraphComplement of G1;
  let e1,e2,v,w be object;
  assume A1: e1 DJoins v,w,G1;
  per cases;
  suppose A2: v <> w;
    e1 Joins v,w,G1 by A1, GLIB_000:16;
    then v is Vertex of G1 & w is Vertex of G1 by GLIB_000:13;
    hence thesis by A1, A2, Th80;
  end;
  suppose v = w;
    hence thesis by GLIB_000:136;
  end;
end;
