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 Th9:
  a is limit_ordinal & 0 in b implies exp(a,b) is limit_ordinal proof assume
A1: a is limit_ordinal & 0 in b;
    per cases by ORDINAL3:8;
    suppose a = 0;
      hence thesis by A1,ORDINAL4:20;
    end;
    suppose
A2:   0 in a;
      defpred P[Ordinal] means 0 in $1 implies exp(a,$1) is limit_ordinal;
A3:   P[0];
A4:   P[c] implies P[succ c]
      proof
        exp(a, succ c) = a*^exp(a,c) by ORDINAL2:44;
        hence thesis by A1,ORDINAL3:40;
      end;
A5:   c <> 0 & c is limit_ordinal & (for b st b in c holds P[b]) implies P[c]
      proof assume that
A6:     c <> 0 & c is limit_ordinal and
A7:     for b st b in c holds P[b];
        deffunc F(Ordinal) = exp(a,$1);
        consider f such that
A8:     dom f = c & for b st b in c holds f.b = F(b) from ORDINAL2:sch 3;
A9:     exp(a,c) = lim f by A6,A8,ORDINAL2:45;
        f is non-decreasing by A2,A8,Th8; then
        Union f is_limes_of f by A6,A8,Th6; then
A10:     exp(a,c) = Union f by A9,ORDINAL2:def 10;
        for d st d in exp(a,c) holds succ d in exp(a,c)
        proof let d; assume d in exp(a,c); then
          consider b being object such that
A11:       b in dom f & d in f.b by A10,CARD_5:2;
          reconsider b as Ordinal by A11;
A12:       succ b in dom f by A6,A8,A11,ORDINAL1:28; then
A13:       f.b = F(b) & f.succ b = F(succ b) & P[succ b] by A7,A8,A11;
          F(b) c= F(succ b) by A2,ORDINAL3:1,ORDINAL4:27; then
          succ d in f.succ b by A13,A11,ORDINAL1:28,ORDINAL3:8;
          hence succ d in exp(a,c) by A10,A12,CARD_5:2;
        end;
        hence P[c] by ORDINAL1:28;
      end;
      P[c] from ORDINAL2:sch 1(A3,A4,A5);
      hence thesis by A1;
    end;
  end;
