
theorem Th14:
  for G1 being _Graph, G2 being DLGraphComplement of G1
  holds the_Edges_of G1 = G1.loops() iff G2 is Dcomplete
proof
  let G1 be _Graph, G2 be DLGraphComplement of G1;
  hereby
    assume A1: the_Edges_of G1 = G1.loops();
    now
      let v2,w2 be Vertex of G2;
      reconsider v1=v2,w1=w2 as Vertex of G1 by GLIB_012:def 6;
      assume A2: v2 <> w2;
      not ex e1 being object st e1 DJoins v1,w1,G1
      proof
        given e1 being object such that
          A3: e1 DJoins v1,w1,G1;
        e1 Joins v1,w1,G1 by A3, GLIB_000:16;
        then not e1 in G1.loops() by A2, GLIB_009:46;
        hence contradiction by A1, A3, GLIB_000:def 14;
      end;
      hence ex e2 being object st e2 DJoins v2,w2,G2 by GLIB_012:def 6;
    end;
    hence G2 is Dcomplete;
  end;
  assume A4: G2 is Dcomplete;
  now
    let e be object;
    set v1 = (the_Source_of G1).e, w1 = (the_Target_of G1).e;
    assume e in the_Edges_of G1;
    then A5: e DJoins v1,w1,G1 by GLIB_000:def 14;
    then e Joins v1,w1,G1 by GLIB_000:16;
    then reconsider v1,w1 as Vertex of G1 by GLIB_000:13;
    reconsider v2=v1, w2=w1 as Vertex of G2 by GLIB_012:def 6;
    v1 = w1
    proof
      assume A6: v1 <> w1;
      not ex e2 being object st e2 DJoins v2,w2,G2 by A5, GLIB_012:def 6;
      hence contradiction by A4, A6;
    end;
    hence e in G1.loops() by A5, GLIB_009:45;
  end;
  then the_Edges_of G1 c= G1.loops() by TARSKI:def 3;
  hence thesis by XBOOLE_0:def 10;
end;
