reserve G for _Graph;

theorem Th16:
  for H being removeDParallelEdges of G
  holds VertexDomRel(H) = VertexDomRel(G)
proof
  let H be removeDParallelEdges of G;
  consider E being RepDEdgeSelection of G such that
    A1: H is inducedSubgraph of G, the_Vertices_of G, E by GLIB_009:def 8;
  A2: VertexDomRel(H) c= VertexDomRel(G) by Th15;
  now
    let v,w be object;
    assume [v,w] in VertexDomRel(G);
    then consider e0 being object such that
      A3: e0 DJoins v,w,G by Th1;
    the_Edges_of G = G.edgesBetween(the_Vertices_of G) &
      the_Vertices_of G c= the_Vertices_of G by GLIB_000:34;
    then A4: the_Edges_of H = E by A1, GLIB_000:def 37;
    consider e being object such that
      A5: e DJoins v,w,G & e in E and
      for e9 being object st e9 DJoins v,w,G & e9 in E holds e9 = e
      by A3, GLIB_009:def 6;
    v is set & w is set by TARSKI:1;
    then e DJoins v,w,H by A4, A5, GLIB_000:73;
    hence [v,w] in VertexDomRel(H) by Th1;
  end;
  then VertexDomRel(G) c= VertexDomRel(H) by RELAT_1:def 3;
  hence thesis by A2, XBOOLE_0:def 10;
end;
