
theorem Th77:
  for a, b being Ordinal holds dom CantorNF a c= dom CantorNF(a(+)b)
proof
  let a, b be Ordinal;
  set E1 = omega -exponent CantorNF a, E2 = omega -exponent CantorNF b;
  set C0 = CantorNF(a(+)b);
  A1: dom CantorNF a = card dom E1 by Def1
    .= card rng E1 by CARD_1:70;
  card rng E1 c= card(rng E1 \/ rng E2) by XBOOLE_1:7, CARD_1:11;
  then card rng E1 c= card rng(omega -exponent C0) by Th76;
  then dom CantorNF a c= card dom(omega -exponent C0) by A1, CARD_1:70;
  hence dom CantorNF a c= dom C0 by Def1;
end;
