
theorem Th92:
  for G1, G2 being _Graph, F being one-to-one PGraphMapping of G1, G2
  st F is total holds card(G1.loops()) c= card(G2.loops())
proof
  let G1, G2 be _Graph, F be one-to-one PGraphMapping of G1, G2;
  assume A1: F is total;
  then reconsider F as non empty one-to-one PGraphMapping of G1, G2;
  A2: (dom F).loops() = G1.loops() by A1, Th55, GLIB_009:50;
  A3: card (rng F).loops() c= card G2.loops() by GLIB_009:48, CARD_1:11;
  card (dom F).loops() c= card (rng F).loops() by Lm4;
  hence thesis by A2, A3, XBOOLE_1:1;
end;
