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 Th21:
  a c= b & 0 in c implies c |^|^ a c= c |^|^ b proof assume that
A1: a c= b and
A2: 0 in c;
    succ 0 c= c & succ 0 = 0+1 by A2,ORDINAL1:21; then
A3: 1 c< c or 1 = c;
    per cases by A3,ORDINAL1:11;
    suppose c = 1; then
      c |^|^ a = 1 & c |^|^ b = 1 by Th17;
      hence thesis;
    end;
    suppose
A4:   1 in c;
      defpred H[Ordinal] means
      for a,b st a c= b & b c= $1 holds c |^|^ a c= c |^|^ b;
A5:   H[0] proof let a,b;
        assume
A6:     a c= b & b c= 0; then
        b = {};
        hence thesis by A6;
      end;
A7:   now let d such that
A8:     H[d];
        c |^|^ (succ d) = exp(c, c |^|^ d) by Th14; then
A9:     c |^|^ d c= c |^|^ (succ d) by A4,ORDINAL4:32;
        thus H[succ d] proof let a,b such that
A10:       a c= b & b c= succ d;
A11:       a c= succ d by A10;
          per cases by A10,A11,Th1;
          suppose b c= d;
            hence thesis by A8,A10;
          end;
          suppose b = succ d & a = succ d;
            hence thesis;
          end;
          suppose
A12:         b = succ d & a c= d; then
            c |^|^ a c= c |^|^ d by A8;
            hence thesis by A9,A12;
          end;
        end;
      end;
A13:   now let d such that
A14:     d <> 0 & d is limit_ordinal and
A15:     for e st e in d holds H[e];
        deffunc E(Ordinal) = c|^|^$1;
        consider f being Ordinal-Sequence such that
A16:     dom f = d & for e st e in d holds f.e = E(e) from ORDINAL2:sch 3;
        f is non-decreasing proof let a,b; assume
A17:       a in b & b in dom f; then
          a c= b & H[b] by A15,A16,ORDINAL1:def 2; then
          c |^|^ a c= c |^|^ b & a in d & f.b = E(b) by A16,A17,ORDINAL1:12;
          hence thesis by A16;
        end; then
A18:     Union f is_limes_of f by A14,A16,Th6;
        c|^|^d = lim f by A14,A16,Th15; then
A19:     c|^|^d = Union f by A18,ORDINAL2:def 10;
        thus H[d]
        proof
          let a,b; assume
A20:       a c= b & b c= d; then
A21:       (b c< d or b = d) & (a c< b or a = b);
          per cases by A21,ORDINAL1:11;
          suppose b in d;
            hence E(a) c= E(b) by A20,A15;
          end;
          suppose
A22:         b = d & a in d; then
            f.a in rng f & f.a = E(a) by A16,FUNCT_1:def 3;
            hence E(a) c= E(b) by A19,A22,ZFMISC_1:74;
          end;
          suppose b = d & a = d;
            hence E(a) c= E(b);
          end;
        end;
      end;
      for b holds H[b] from ORDINAL2:sch 1(A5,A7,A13);
      hence thesis by A1;
    end;
  end;
