
theorem Th112:
  for G1 being _Graph, G2 being removeDParallelEdges of G1
  for v1 being Vertex of G1, v2 being Vertex of G2
  st v1 = v2 holds v1 is isolated iff v2 is isolated
proof
  let G1 be _Graph, G2 be removeDParallelEdges of G1;
  set G3 = the removeParallelEdges of G2;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2;
  reconsider v3 = v2 as Vertex of G3 by GLIB_000:def 33;
  G3 is removeParallelEdges of G1 by Th95;
  then v1 is isolated iff v3 is isolated by A1, Th111;
  hence thesis by Th111;
end;
