reserve
  a,b,c,d,e for Ordinal,
  m,n for Nat,
  f for Ordinal-Sequence,
  x for object;
reserve S,S1,S2 for Sequence;

theorem Th33:
  ex b st a in b & b is epsilon
  proof
    deffunc phi(Ordinal) = exp(omega, $1);
A1: for a,b st a in b holds phi(a) in phi(b) by ORDINAL4:24;
A2: now let a such that
A3:   a is non empty limit_ordinal;
      let phi being Ordinal-Sequence such that
A4:   dom phi = a & for b st b in a holds phi.b = phi(b);
      phi is non-decreasing
      proof
        let b,c; assume
A5:     b in c & c in dom phi; then
        phi.b = phi(b) & phi.c = phi(c) by A4,ORDINAL1:10; then
        phi.b in phi.c by A5,ORDINAL4:24;
        hence thesis by ORDINAL1:def 2;
      end; then
      Union phi is_limes_of phi & phi(a) = lim phi by A3,A4,Th6,ORDINAL2:45;
      hence phi(a) is_limes_of phi by ORDINAL2:def 10;
    end;
    consider b such that
A6: a in b & phi(b) = b from CriticalNumber3(A1,A2);
    take b; thus a in b by A6;
    thus exp(omega, b) = b by A6;
  end;
