
theorem Th102:
  for a, b being Ordinal st omega -exponent a in omega -exponent b
  holds a in exp(omega, omega -exponent b)
proof
  defpred P[non empty Ordinal] means for b being Ordinal
    st omega -exponent $1 in omega -exponent b
    holds $1 in exp(omega, omega -exponent b);
  A1: for c being Ordinal, n being non zero Nat holds P[n*^exp(omega,c)]
  proof
    let c be Ordinal, n be non zero Nat, b be Ordinal;
    assume A2: omega -exponent(n*^exp(omega,c)) in omega -exponent b;
    0 in n & n in omega by XBOOLE_1:61, ORDINAL1:11, ORDINAL1:def 12;
    then c in omega -exponent b by A2, ORDINAL5:58;
    then exp(omega,c) in exp(omega, omega -exponent b) by ORDINAL4:24;
    hence thesis by Th42;
  end;
  A3: for c being Ordinal, d being non empty Ordinal, n being non zero Nat
    st P[d] & not c in rng(omega -exponent CantorNF d)
    holds P[d (+) n*^exp(omega,c)]
  proof
    let c be Ordinal, d be non empty Ordinal, n be non zero Nat;
    assume A4: P[d] & not c in rng(omega -exponent CantorNF d);
    let b be Ordinal;
    assume omega -exponent(d (+) n*^exp(omega,c)) in omega -exponent b;
    then omega -exponent d \/ omega -exponent(n*^exp(omega,c))
      in omega -exponent b by Th100;
    then omega -exponent d in omega -exponent b &
      omega -exponent(n*^exp(omega,c)) in omega -exponent b
      by XBOOLE_1:7, ORDINAL1:12;
    then d in exp(omega, omega -exponent b) &
      n*^exp(omega,c) in exp(omega, omega -exponent b) by A1, A4;
    hence d (+) n*^exp(omega,c) in exp(omega, omega -exponent b) by Th101;
  end;
  A5: for a being non empty Ordinal holds P[a] from OrdinalCNFIndA(A1,A3);
  let a, b be Ordinal;
  per cases;
  suppose a <> {};
    hence thesis by A5;
  end;
  suppose A6: a = {};
    assume omega -exponent a in omega -exponent b;
    thus thesis by A6, XBOOLE_1:61, ORDINAL1:11;
  end;
end;
