
theorem Th57:
  for G1 being _Graph, G2 being DLGraphComplement of G1
  st the_Edges_of G1 = G1.loops() holds G2 is complete
proof
  let G1 be _Graph, G2 be DLGraphComplement of G1;
  assume A1: the_Edges_of G1 = G1.loops();
  now
    let v,w be Vertex of G2;
    A2: v is Vertex of G1 & w is Vertex of G1 by Def6;
    assume A3: v <> w;
    not ex e1 being object st e1 DJoins v,w,G1
    proof
      given e1 being object such that
        A4: e1 DJoins v,w,G1;
      e1 Joins v,w,G1 by A4, GLIB_000:16;
      hence contradiction by A1, A3, A4, GLIB_009:46, GLIB_000:def 14;
    end;
    then consider e2 being object such that
      A5: e2 DJoins v,w,G2 by A2, Def6;
    e2 Joins v,w,G2 by A5, GLIB_000:16;
    hence v,w are_adjacent by CHORD:def 3;
  end;
  hence thesis by CHORD:def 6;
end;
