
theorem Th96:
  for G1 being _Graph, G2 being removeDParallelEdges of G1
  ex G3 being removeParallelEdges of G1
  st G3 is removeParallelEdges of G2
proof
  let G1 be _Graph, G2 be removeDParallelEdges of G1;
  consider E1 being RepDEdgeSelection of G1 such that
    A1: G2 is inducedSubgraph of G1, the_Vertices_of G1, E1 by Def8;
  consider E2 being RepEdgeSelection of G1 such that
    A2: E2 c= E1 by Th72;
  set G3 = the inducedSubgraph of G1, the_Vertices_of G1, E2;
  reconsider G3 as removeParallelEdges of G1 by Def7;
  take G3;
  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: G3 is Subgraph of G2 by A1, A2, GLIB_000:46;
  A5: the_Vertices_of G2 c= the_Vertices_of G2 &
    the_Edges_of G2 = G2.edgesBetween(the_Vertices_of G2) by GLIB_000:34;
  the_Vertices_of G2 = the_Vertices_of G1 by A1, A3, GLIB_000:def 37;
  then A6: the_Vertices_of G3 = the_Vertices_of G2 by A3, GLIB_000:def 37;
  A7: E2 c= the_Edges_of G2 by A1, A2, A3, GLIB_000:def 37;
  the_Edges_of G3 = E2 by A3, GLIB_000:def 37;
  then A8: G3 is inducedSubgraph of G2, the_Vertices_of G2, E2
    by A4, A5, A6, A7, GLIB_000:def 37;
  E2 is RepEdgeSelection of G2 by A7, Th78;
  hence thesis by A8, Def7;
end;
