
theorem Th85:
  for a, b being Ordinal
  st a <> 0 implies omega -exponent b in omega -exponent last CantorNF a
  holds a (+) b = a +^ b
proof
  let a, b be Ordinal;
  assume A1: a <> 0 implies omega-exponent b in omega-exponent last CantorNF a;
  per cases;
  suppose a = 0;
    then a (+) b = b & a +^ b = b by Th82, ORDINAL2:30;
    hence thesis;
  end;
  suppose b = 0;
    then a (+) b = a & a +^ b = a by Th82, ORDINAL2:27;
    hence thesis;
  end;
  suppose A2: a <> 0 & b <> 0;
    then omega-exponent Sum^ CantorNF b in omega-exponent last CantorNF a
      by A1;
    then omega -exponent((CantorNF b).0) in omega -exponent last CantorNF a
      by Th44;
    then A3: CantorNF a ^ CantorNF b is Cantor-normal-form by A2, Th33;
    thus a (+) b = Sum^ CantorNF a (+) Sum^ CantorNF b
      .= a +^ b by A3, Th84;
  end;
end;
