
theorem
  for G1, G2 being _Graph, F being non empty one-to-one PGraphMapping of G1, G2
  holds card((dom F).loops()) = card((rng F).loops())
proof
  let G1, G2 be _Graph, F be non empty one-to-one PGraphMapping of G1, G2;
  reconsider F2 = F" as non empty one-to-one PGraphMapping of G2, G1;
  A1: card((dom F).loops()) c= card((rng F).loops()) &
    card((dom F2).loops()) c= card((rng F2).loops()) by Lm4;
  dom F2 = rng F & rng F2 = dom F by Th81;
  hence thesis by A1, XBOOLE_0:def 10;
end;
