
theorem Th39:
  for A being non empty Cantor-normal-form Ordinal-Sequence
  for b being Ordinal, n being non zero Nat
  st omega -exponent(A.0) in b
  holds <% n*^exp(omega,b) %> ^ A 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: omega -exponent(A.0) in b;
  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(A.0) in omega -exponent(n*^exp(omega,b))
    by A1, ORDINAL5:58;
  then omega -exponent(A.0) in omega -exponent last({}^<%n*^exp(omega,b)%>)
    by AFINSQ_1:92;
  hence thesis by Th33;
end;
