
theorem Th84:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2 st F is isomorphism
  holds G1.order() = G2.order() & G1.size() = G2.size()
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is isomorphism;
  then reconsider F as one-to-one PGraphMapping of G1, G2;
  A2: G1.order() c= G2.order() & G1.size() c= G2.size() by A1, Th45;
  F" is isomorphism by A1, Th75;
  then G2.order() c= G1.order() & G2.size() c= G1.size() by Th45;
  hence thesis by A2, XBOOLE_0:def 10;
end;
