
theorem
  for G1 being non-Dmulti _Graph, G2 being _Graph
  holds G1 == G2 iff G2 is removeDParallelEdges of G1
proof
  let G1 be non-Dmulti _Graph, G2 be _Graph;
  hereby
    assume A1: G1 == G2;
    set E = the RepDEdgeSelection of G1;
    G1 is inducedSubgraph of G1, the_Vertices_of G1 by GLIB_000:100;
    then G1 is inducedSubgraph of G1, the_Vertices_of G1, the_Edges_of G1
      by GLIB_000:34;
    then G2 is inducedSubgraph of G1, the_Vertices_of G1, the_Edges_of G1
      by A1, GLIB_000:101;
    then G2 is inducedSubgraph of G1, the_Vertices_of G1, E by Th76;
    hence G2 is removeDParallelEdges of G1 by Def8;
  end;
  assume G2 is removeDParallelEdges of G1;
  then consider E being RepDEdgeSelection of G1 such that
    A2: G2 is inducedSubgraph of G1, the_Vertices_of G1, E by Def8;
  G2 is inducedSubgraph of G1, the_Vertices_of G1, the_Edges_of G1
    by A2, Th76;
  then A3: G2 is inducedSubgraph of G1, the_Vertices_of G1 by GLIB_000:34;
  G1 is inducedSubgraph of G1, the_Vertices_of G1 by GLIB_000:100;
  hence G1 == G2 by A3, GLIB_000:93;
end;
