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;
reserve A,B for Cantor-normal-form Ordinal-Sequence;

theorem
  ex A being Cantor-normal-form Ordinal-Sequence st a = Sum^ A
  proof
    defpred P[Ordinal] means 0 in $1 implies
    ex A being non empty Cantor-normal-form Ordinal-Sequence st $1 = Sum^ A;
A1: for a st for b st b in a holds P[b] holds P[a] proof let a such that
A2:   for b st b in a holds P[b] and
A3:   0 in a;
      consider n,b such that
A4:   a = n*^exp(omega, omega-exponent(a))+^b & 0 in Segm n &
      b in exp(omega, omega-exponent(a)) by A3,Th62;
      reconsider s = n*^exp(omega, omega-exponent(a)) as
      Cantor-component Ordinal by A4,Def9;
      set c = omega-exponent(a);
A5:   exp(omega, c) c= a by A3,Def10;
      per cases by ORDINAL3:8;
      suppose
A6:     b = 0;
        reconsider A = <%n*^exp(omega, omega-exponent(a))%> as
        non empty Cantor-normal-form Ordinal-Sequence by A4,Th65;
        take A;
        thus a = n*^exp(omega, omega-exponent(a)) by A4,A6,ORDINAL2:27
        .= Sum^ A by Th53;
      end;
      suppose
        0 in b; then
        consider A being non empty Cantor-normal-form Ordinal-Sequence
        such that
A7:     b = Sum^ A by A5,A2,A4;
A8:     A.0 in exp(omega, omega-exponent(a)) by A4,A7,Th56,ORDINAL1:12;
        0 in dom A by ORDINAL3:8; then
        A.0 is Cantor-component Ordinal by Def11; then
        0 in A.0 by ORDINAL3:8; then
        exp(omega, omega-exponent(A.0)) c= A.0 by Def10; then
A9:     exp(omega, omega-exponent(A.0)) in exp(omega, omega-exponent(a))
        by A8,ORDINAL1:12;
        n in omega by ORDINAL1:def 12; then
        omega-exponent(s) = omega-exponent(a) by A4,Th58; then
        reconsider B = <%s%>^A as non empty Cantor-normal-form Ordinal-Sequence
        by A9,Th68,Th12;
        take B;
        thus a = Sum^ B by A4,A7,Th55;
      end;
    end;
A10: P[b] from ORDINAL1:sch 2(A1);
    per cases by ORDINAL3:8;
    suppose
A11:   a = 0;
      reconsider A = {} as Cantor-normal-form Ordinal-Sequence;
      take A;
      thus thesis by A11,Th52;
    end;
    suppose 0 in a; then
      ex A being non empty Cantor-normal-form Ordinal-Sequence st a = Sum^ A
      by A10;
      hence thesis;
    end;
  end;
