
theorem Th17:
  for G1 being _Graph, G2 being DGraphComplement of G1
  holds G1 is Dcomplete iff G2 is edgeless
proof
  let G1 be _Graph, G2 be DGraphComplement of G1;
  consider G9 being DLGraphComplement of G1 such that
    A1: G2 is removeLoops of G9 by GLIB_012:def 8;
  hereby
    assume G1 is Dcomplete;
    then the_Edges_of G9 = G9.loops() by Th15;
    hence G2 is edgeless by A1;
  end;
  assume G2 is edgeless;
  then {} = the_Edges_of G2
    .= the_Edges_of G9 \ G9.loops() by A1, GLIB_000:53;
  then the_Edges_of G9 c= G9.loops() by XBOOLE_1:37;
  hence G1 is Dcomplete by Th15, XBOOLE_0:def 10;
end;
