reserve G for _Graph;

theorem
  for H being SimpleGraph of G
  holds VertexAdjSymRel(H) = VertexAdjSymRel(G) \ id the_Vertices_of G
proof
  let H be SimpleGraph of G;
  consider G9 being removeParallelEdges of G such that
    A1: H is removeLoops of G9 by GLIB_009:119;
  A2: the_Vertices_of G9 = the_Vertices_of G by GLIB_000:def 33;
  thus VertexAdjSymRel(H)
     = VertexAdjSymRel(G9) \ id the_Vertices_of G9 by A1, Th47
    .= VertexAdjSymRel(G) \ id the_Vertices_of G by A2, Th46;
end;
