
theorem Th37:
  for A being non empty Cantor-normal-form Ordinal-Sequence
  for b being Ordinal, n being non zero Nat
  st b in omega -exponent last A
  holds A ^ <% n*^exp(omega,b) %> is Cantor-normal-form
proof
  let A be non empty Cantor-normal-form Ordinal-Sequence;
  let b be Ordinal, n be non zero Nat;
  assume A1: b in omega -exponent last A;
  0 c< n by XBOOLE_1:2, XBOOLE_0:def 8;
  then 0 in n & n in omega by ORDINAL1:11, ORDINAL1:def 12;
  then omega-exponent(<% n *^ exp(omega,b) %>.0) in omega-exponent last A
    by A1, ORDINAL5:58;
  hence thesis by Th33;
end;
