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
  0 in a & omega c= b implies a |^|^ b = a |^|^ omega proof assume
A1: 0 in a;
    assume omega c= b; then
A2: b = omega+^(b-^omega) by ORDINAL3:def 5;
    defpred P[Ordinal] means a |^|^ (omega+^$1) = a |^|^ omega;
A3: P[0] by ORDINAL2:27;
A4: P[c] implies P[succ c]
    proof
      assume P[c]; then
A5:   exp(a, a |^|^ (omega+^c)) = a |^|^ omega by A1,Th31;
      thus a |^|^ (omega+^succ c) = a |^|^ succ (omega+^c) by ORDINAL2:28
      .= a |^|^ omega by A5,Th14;
    end;
A6: c <> 0 & c is limit_ordinal & (for b st b in c holds P[b]) implies P[c]
    proof assume
A7:   c <> 0 & c is limit_ordinal;
      assume
A8:   for b st b in c holds P[b];
      deffunc F(Ordinal) = a |^|^ $1;
      consider f such that
A9:   dom f = omega+^c & for b st b in omega+^c holds f.b = F(b)
      from ORDINAL2:sch 3;
      omega+^c <> {} & omega+^c is limit_ordinal by A7,ORDINAL3:26,29; then
A10:   a|^|^(omega+^c) = lim f by A9,Th15;
      now a c= a|^|^omega by A1,Th23;
        hence a|^|^omega <> {} by A1;
        let B,C be Ordinal;
        assume
A11:     B in a|^|^omega & a|^|^omega in C;
        take D = omega;
        omega+^({} qua Ordinal) = omega & {} in c
        by A7,ORDINAL2:27,ORDINAL3:8;
        hence D in dom f by A9,ORDINAL2:32;
        let E be Ordinal;
        assume
A12:     D c= E & E in dom f; then
        E-^D in (omega+^c)-^omega by A9,ORDINAL3:53; then
        E-^D in c by ORDINAL3:52; then
        P[E-^D] by A8; then
        a|^|^omega = a|^|^E by A12,ORDINAL3:def 5;
        hence B in f.E & f.E in C by A9,A11,A12;
      end; then
      a|^|^omega is_limes_of f by ORDINAL2:def 9;
      hence P[c] by A10,ORDINAL2:def 10;
    end;
    P[c] from ORDINAL2:sch 1(A3,A4,A6);
    hence a |^|^ b = a |^|^ omega by A2;
  end;
