
theorem Th21:
  for a, b being Ordinal st a in b holds b -exponent a = 0
proof
  let a, b be Ordinal;
  assume A1: a in b;
  per cases;
  suppose 0 in a;
    then 0 = b -exponent(a *^ exp(b,0)) by A1, ORDINAL5:58
      .= b -exponent(a *^ 1) by ORDINAL2:43
      .= b -exponent a by ORDINAL2:39;
    hence thesis;
  end;
  suppose not 0 in a;
    hence thesis by ORDINAL5:def 10;
  end;
end;
