
theorem
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is isomorphism holds
    (G1 is _trivial iff G2 is _trivial) &
    (G1 is loopless iff G2 is loopless) &
    (G1 is edgeless iff G2 is edgeless) &
    (G1 is non-multi iff G2 is non-multi) &
    (G1 is simple iff G2 is simple) &
    (G1 is _finite iff G2 is _finite) &
    (G1 is complete iff G2 is complete)
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;
  F" is isomorphism by A1, Th75;
  hence thesis by A1, Th35, Th49, Th52;
end;
