
theorem Th95:
  for G1 being _Graph, G2 being removeDParallelEdges of G1
  for G3 being removeParallelEdges of G2
  holds G3 is removeParallelEdges of G1
proof
  let G1 be _Graph, G2 be removeDParallelEdges of G1;
  let G3 be removeParallelEdges of G2;
  A1: G3 is Subgraph of G1 by GLIB_000:43;
  consider E1 being RepDEdgeSelection of G1 such that
    A2: G2 is inducedSubgraph of G1, the_Vertices_of G1, E1 by Def8;
  A3: the_Vertices_of G1 c= the_Vertices_of G1 &
    the_Edges_of G1 = G1.edgesBetween(the_Vertices_of G1) by GLIB_000:34;
  then A4: the_Vertices_of G2 = the_Vertices_of G1 &
    the_Edges_of G2 = E1 by A2, GLIB_000:def 37;
  consider E2 being RepEdgeSelection of G2 such that
    A5: G3 is inducedSubgraph of G2, the_Vertices_of G2, E2 by Def7;
  the_Vertices_of G2 c= the_Vertices_of G2 &
    the_Edges_of G2 = G2.edgesBetween(the_Vertices_of G2) by GLIB_000:34;
  then A6: the_Vertices_of G3 = the_Vertices_of G2 &
    the_Edges_of G3 = E2 by A5, GLIB_000:def 37;
  E2 c= the_Edges_of G1 by A1, A6, GLIB_000:def 32;
  then A7: G3 is inducedSubgraph of G1, the_Vertices_of G1, E2
    by A1, A3, A4, A6, GLIB_000:def 37;
  E2 is RepEdgeSelection of G1 by A2, Th84;
  hence thesis by A7, Def7;
end;
