reserve G for _Graph;

theorem Th46:
  for H being removeParallelEdges of G
  holds VertexAdjSymRel(H) = VertexAdjSymRel(G)
proof
  let H be removeParallelEdges of G;
  consider E being RepEdgeSelection of G such that
    A1: H is inducedSubgraph of G, the_Vertices_of G, E by GLIB_009:def 7;
  A2: VertexAdjSymRel(H) c= VertexAdjSymRel(G) by Th45;
  now
    let v,w be object;
    assume [v,w] in VertexAdjSymRel(G);
    then consider e0 being object such that
      A3: e0 Joins v,w,G by Th32;
    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 Joins v,w,G & e in E and
      for e9 being object st e9 Joins v,w,G & e9 in E holds e9 = e
      by A3, GLIB_009:def 5;
    v is set & w is set by TARSKI:1;
    then e Joins v,w,H by A4, A5, GLIB_000:73;
    hence [v,w] in VertexAdjSymRel(H) by Th32;
  end;
  then VertexAdjSymRel(G) c= VertexAdjSymRel(H) by RELAT_1:def 3;
  hence thesis by A2, XBOOLE_0:def 10;
end;
