
theorem
  for a, b being Ordinal st omega -exponent a in omega -exponent b
  holds b -^ a = b
proof
  let a, b be Ordinal;
  assume A1: omega -exponent a in omega -exponent b;
  per cases;
  suppose a = 0;
    hence thesis by ORDINAL3:56;
  end;
  suppose A2: a <> 0;
    A3: 1 in omega & 0 in b by A1, ORDINAL5:def 10;
    then omega *^ a c= b by A1, A2, Th103;
    then A4: a +^ b = b by Th30;
    A5: a in exp(omega, omega -exponent b) by A1, Th102;
    exp(omega, omega -exponent b) c= b by A3, ORDINAL5:def 10;
    then a c= b by A5, ORDINAL1:def 2;
    hence thesis by A4, ORDINAL3:def 5;
  end;
end;
