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 Th44:
  a in b implies epsilon_a in epsilon_b
  proof
    defpred P[Ordinal] means 1 in epsilon_$1 &
    for a,b st a in b & b c= $1 holds epsilon_a in epsilon_b;
    omega in epsilon_0 by Th26,Th41; then
A1: P[0] by ORDINAL1:10;
A2: P[c] implies P[succ c] proof assume
A3:   P[c];
A4:   epsilon_succ c = (epsilon_c) |^|^ omega by Th42; then
A5:   epsilon_c in epsilon_succ c by A3,Th26;
      hence 1 in epsilon_succ c by A3,ORDINAL1:10;
      let a,b; assume
A6:   a in b & b c= succ c; then
      a c= c by ORDINAL1:22; then
A7:   b = succ c & (a = c or a c< c) or b c< succ c by A6;
      per cases by A7,ORDINAL1:11;
      suppose b in succ c; then
        b c= c by ORDINAL1:22;
        hence thesis by A3,A6;
      end;
      suppose
A8:     b = succ c & a in c; then
        epsilon_a in epsilon_c by A3;
        hence thesis by A5,A8,ORDINAL1:10;
      end;
      suppose
        b = succ c & a = c;
        hence thesis by A3,A4,Th26;
      end;
    end;
A9: c <> 0 & c is limit_ordinal & (for b st b in c holds P[b]) implies P[c]
    proof assume that
A10:   c <> 0 & c is limit_ordinal and
A11:   for b st b in c holds P[b];
      deffunc F(Ordinal) = epsilon_$1;
      consider f such that
A12:   dom f = c & for b st b in c holds f.b = F(b) from ORDINAL2:sch 3;
      f is increasing proof let a,b; assume
A13:     a in b & b in dom f; then
        a in c by A12,ORDINAL1:10; then
        f.a = F(a) & f.b = F(b) & P[b] by A11,A12,A13;
        hence thesis by A13;
      end; then
      Union f is_limes_of f by A10,A12,Th6; then
A14:   Union f = lim f by ORDINAL2:def 10 .= epsilon_c by A10,A12,Th43;
A15:   0 in c by A10,ORDINAL3:8; then
      1 in epsilon_0 & f.0 = F(0) by A11,A12;
      hence 1 in epsilon_c by A12,A14,A15,CARD_5:2;
      let a,b; assume
A16:   a in b & b c= c; then
A17:   b = c or b c< c;
      per cases by A17,ORDINAL1:11;
      suppose b in c;
        hence epsilon_a in epsilon_b by A11,A16;
      end;
      suppose
A18:     b = c;
        succ a in c & a in succ a by A10,A16,ORDINAL1:6,28; then
A19:     F(a) in F(succ a) & F(succ a) = f.succ a & f.succ a in rng f
        by A11,A12,FUNCT_1:def 3; then
        f.succ a c= Union f by ZFMISC_1:74;
        hence thesis by A14,A18,A19;
      end;
    end;
    P[c] from ORDINAL2:sch 1(A1,A2,A9);
    hence thesis;
  end;
