
theorem Th6:
  for G1 being _Graph, G2 being removeDParallelEdges of G1
  holds G1 is Dcomplete iff G2 is Dcomplete
proof
  let G1 be _Graph, G2 be removeDParallelEdges 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 v <> w;
      then consider e0 being object such that
        A3: e0 DJoins v,w,G1 by A1, A2;
      consider E being RepDEdgeSelection of G1 such that
        A4: G2 is inducedSubgraph of G1,the_Vertices_of G1,E by GLIB_009:def 8;
      consider e being object such that
        A5: e DJoins v,w,G1 & e in E and
        for e9 being object st e9 DJoins v,w,G1 & e9 in E holds e9 = e
        by A3, GLIB_009:def 6;
      take e;
      the_Vertices_of G1 c= the_Vertices_of G1;
      then the_Vertices_of G1 is non empty Subset of the_Vertices_of G1 &
        G1.edgesBetween(the_Vertices_of G1) = the_Edges_of G1 by GLIB_000:34;
      then the_Edges_of G2 = E by A4, GLIB_000:def 37;
      hence e DJoins v,w,G2 by A5, 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;
