
theorem Th100:
  for a, b being Ordinal
  holds omega -exponent(a (+) b) = omega -exponent a \/ omega -exponent b
proof
  let a, b be Ordinal;
  per cases;
  suppose A1: a (+) b <> {};
    omega -exponent a c= omega -exponent (a (+) b) &
      omega -exponent b c= omega -exponent (a (+) b) by Th22, Th99;
    then A2: omega -exponent a \/ omega -exponent b c= omega -exponent(a (+) b)
      by XBOOLE_1:8;
    set E1 = omega -exponent CantorNF a, E2 = omega -exponent CantorNF b;
    set C0 = CantorNF(a(+)b);
    0 c< dom C0 by A1, XBOOLE_1:2, XBOOLE_0:def 8;
    then A3: 0 in dom C0 by ORDINAL1:11;
    then 0 in dom(omega -exponent C0) by Def1;
    then (omega -exponent C0).0 in rng(omega -exponent C0) by FUNCT_1:3;
    then (omega -exponent C0).0 in rng E1 \/ rng E2 by Th76;
    then per cases by XBOOLE_0:def 3;
    suppose A4: (omega -exponent C0).0 in rng E1;
      then omega -exponent(C0.0) in rng E1 by A3, Def1;
      then A5: omega -exponent(C0.0) = E1.0 by Th95;
      E1 <> {} by A4;
      then 0 c< dom E1 by XBOOLE_1:2, XBOOLE_0:def 8;
      then 0 in dom E1 by ORDINAL1:11;
      then A6: 0 in dom CantorNF a by Def1;
      omega -exponent(a (+) b) = omega -exponent Sum^ C0
        .= omega -exponent(C0.0) by Th44
        .= omega -exponent((CantorNF a).0) by A5, A6, Def1
        .= omega -exponent Sum^ CantorNF a by Th44
        .= omega -exponent a;
      then omega -exponent(a (+) b)
        c= omega -exponent a \/ omega -exponent b by XBOOLE_1:7;
      hence thesis by A2, XBOOLE_0:def 10;
    end;
    suppose A7: (omega -exponent C0).0 in rng E2;
      then omega -exponent(C0.0) in rng E2 by A3, Def1;
      then A8: omega -exponent(C0.0) = E2.0 by Th95;
      E2 <> {} by A7;
      then 0 c< dom E2 by XBOOLE_1:2, XBOOLE_0:def 8;
      then 0 in dom E2 by ORDINAL1:11;
      then A9: 0 in dom CantorNF b by Def1;
      omega -exponent(a (+) b) = omega -exponent Sum^ C0
        .= omega -exponent(C0.0) by Th44
        .= omega -exponent((CantorNF b).0) by A8, A9, Def1
        .= omega -exponent Sum^ CantorNF b by Th44
        .= omega -exponent b;
      then omega -exponent(a (+) b)
        c= omega -exponent a \/ omega -exponent b by XBOOLE_1:7;
      hence thesis by A2, XBOOLE_0:def 10;
    end;
  end;
  suppose a (+) b = {};
    then a = 0 & b = 0;
    hence thesis;
  end;
end;
