
theorem Th5:
  for G1 being _Graph, G2 being removeLoops of G1
  holds G1 is Dcomplete iff G2 is Dcomplete
proof
  let G1 be _Graph, G2 be removeLoops of G1;
  hereby
    assume A1: G1 is Dcomplete;
    now
      let v,w be Vertex of G2;
      A2: v is Vertex of G1 & w is Vertex of G1 by GLIB_000:def 33;
      assume A3: v <> w;
      then consider e being object such that
        A4: e DJoins v,w,G1 by A1, A2;
      take e;
      e Joins v,w,G1 by A4, GLIB_000:16;
      then A5: not e in G1.loops() by A3, GLIB_009:46;
      e in the_Edges_of G1 by A4, GLIB_000:def 14;
      then e in the_Edges_of G1 \ G1.loops() by A5, XBOOLE_0:def 5;
      then e in the_Edges_of G2 by GLIB_000:53;
      hence e DJoins v,w,G2 by A4, GLIB_000:73;
    end;
    hence G2 is Dcomplete;
  end;
  assume A6: G2 is Dcomplete;
  let v,w be Vertex of G1;
  A7: v is Vertex of G2 & w is Vertex of G2 by GLIB_000:def 33;
  assume v <> w;
  then consider e being object such that
    A8: e DJoins v,w,G2 by A6, A7;
  take e;
  thus e DJoins v,w,G1 by A8, GLIB_000:72;
end;
