
theorem
  for G1 being loopless _Graph, G2 being SimpleGraph 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 loopless _Graph, G2 be SimpleGraph of G1;
  consider H being removeParallelEdges of G1 such that
    A1: G2 is removeLoops of H by Th119;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A2: v1 = v2;
  reconsider v3 = v2 as Vertex of H by A1, GLIB_000:def 33;
  G2 == H by A1, Th58;
  then v3 is isolated iff v2 is isolated by GLIB_000:97;
  hence thesis by A2, Th111;
end;
