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 Th31:
  0 in a implies exp(a, a |^|^ omega) = a |^|^ omega proof assume 0 in a; then
    1 = succ 0 & succ 0 c= a by ORDINAL1:21; then
A1: 1 = a or 1 c< a;
    per cases by A1,ORDINAL1:11;
    suppose
A2:   a = 1;
      hence exp(a, a |^|^ omega) = 1 by ORDINAL2:46 .= a |^|^ omega by A2,Th17;
    end;
    suppose
A3:   1 in a;
      deffunc T(Ordinal) = a|^|^$1;
      deffunc E(Ordinal) = exp(a, $1);
      consider T being Ordinal-Sequence such that
A4:   dom T = omega & for a st a in omega holds T.a = T(a) from ORDINAL2:sch 3;
      consider E being Ordinal-Sequence such that
A5:   dom E = a|^|^omega & for b st b in a|^|^omega holds E.b = E(b)
      from ORDINAL2:sch 3;
      0 in Segm 1 by NAT_1:44; then
      0 in a & 0 in omega by A3,ORDINAL1:10; then
A6:   a c= a|^|^omega by Th23;
      E is increasing Ordinal-Sequence by A3,A5,ORDINAL4:25; then
      lim E = exp(a, a|^|^omega) & Union E is_limes_of E
      by A5,A6,Th6,A3,Th30,ORDINAL2:45; then
A7:   exp(a, a|^|^omega) = Union E by ORDINAL2:def 10;
      T is increasing Ordinal-Sequence by A3,A4,Th25; then
      lim T = a|^|^omega & Union T is_limes_of T by A4,Th15,Th6; then
A8:   a|^|^omega = Union T by ORDINAL2:def 10;
      thus exp(a, a |^|^ omega) c= a |^|^ omega
      proof
        let x be Ordinal; assume x in exp(a, a |^|^ omega); then
        consider b being object such that
A9:     b in dom E & x in E.b by A7,CARD_5:2;
        consider c being object such that
A10:     c in dom T & b in T.c by A5,A8,A9,CARD_5:2;
        reconsider b as Ordinal by A9;
        reconsider c as Element of omega by A4,A10;
A11:     exp(a, b) in exp(a, T.c) by A3,A10,ORDINAL4:24;
A12:      succ c in omega by ORDINAL1:def 12;
        then E.b = E(b) & T.c = T(c) & T.succ c = T(succ c) by A9,A4,A5; then
        E.b in T.(succ c) by A11,Th14; then
        x in T.(succ c) by A9,ORDINAL1:10;
        hence thesis by A4,A8,CARD_5:2,A12;
      end;
      thus a |^|^ omega c= exp(a, a |^|^ omega) by A3,ORDINAL4:32;
    end;
  end;
