
theorem Th76:
  for a, b being Ordinal holds rng(omega -exponent CantorNF(a(+)b))
    = rng(omega -exponent CantorNF a) \/ rng(omega -exponent CantorNF b)
proof
  let a, b be Ordinal;
  set E1 = omega -exponent CantorNF a, E2 = omega -exponent CantorNF b;
  set L1 = omega -leading_coeff CantorNF a;
  set L2 = omega -leading_coeff CantorNF b;
  consider C being Cantor-normal-form Ordinal-Sequence such that
    A1: a (+) b = Sum^ C & rng(omega -exponent C) = rng E1 \/ rng E2 and
    for d being object st d in dom C holds
      (omega -exponent(C.d) in rng E1 \ rng E2 implies
        omega -leading_coeff(C.d) = L1.(E1".(omega -exponent(C.d)))) &
      (omega -exponent(C.d) in rng E2 \ rng E1 implies
        omega -leading_coeff(C.d) = L2.(E2".(omega -exponent(C.d)))) &
      (omega -exponent(C.d) in rng E1 /\ rng E2 implies
        omega -leading_coeff(C.d) = L1.(E1".(omega -exponent(C.d))) +
          L2.(E2".(omega -exponent(C.d)))) by Def5;
  thus thesis by A1;
end;
