reserve a,b,c,d for Ordinal;
reserve l for non empty limit_ordinal Ordinal;
reserve u for Element of l;
reserve A for non empty Ordinal;
reserve e for Element of A;
reserve X,Y,x,y,z for set;
reserve n,m for Nat;
reserve f for Ordinal-Sequence;
reserve U,W for Universe;
reserve F,phi for normal Ordinal-Sequence of W;

theorem Th44:
  omega in W implies criticals F is Ordinal-Sequence of W
  proof assume
A1: omega in W;
    set G = criticals F;
A2: dom F = On W & rng F c= On W by FUNCT_2:def 1,RELAT_1:def 19;
A3: rng G c= rng F by Th30; then
A4: rng G c= On W by A2;
    dom G = On W
    proof
      thus dom G c= On W by A2,Th32;
      let a; assume a in On W; then
A5:   a in W by ORDINAL1:def 9;
      defpred P[Ordinal] means $1 in W implies $1 in dom G;
      consider a0 being Ordinal such that
A6:   0-element_of W in a0 & a0 is_a_fixpoint_of F by A1,Th43;
      consider a1 being Ordinal such that
A7:   a1 in dom G & a0 = G.a1 by A6,Th33;
A8:   P[0] by A7,ORDINAL1:12,XBOOLE_1:2;
A9:   for a st P[a] holds P[succ a]
      proof
        let a; assume
A10:     P[a] & succ a in W;
A11:     a c= succ a by ORDINAL3:1; then
        G.a in rng G by A10,CLASSES1:def 1,FUNCT_1:def 3; then
        G.a in W by A4,ORDINAL1:def 9; then
        consider b such that
A12:     G.a in b & b is_a_fixpoint_of F by A1,Th43;
        consider c such that
A13:     c in dom G & b = G.c by A12,Th33;
        a in c by A10,A11,A12,A13,Th23,CLASSES1:def 1; then
        succ a c= c by ORDINAL1:21;
        hence thesis by A13,ORDINAL1:12;
      end;
A14:  for a st a <> 0 & a is limit_ordinal &
      for b st b in a holds P[b] holds P[a]
      proof
        let a such that
A15:    a <> 0 & a is limit_ordinal and
A16:     for b st b in a holds P[b] and
A17:     a in W;
        set X = G.:a;
        card X c= card a & card a c= a by CARD_1:8,67; then
        card X c= a; then
        card X in W by A17,CLASSES1:def 1; then
        card X in On W by ORDINAL1:def 9; then
A18:     card X in card W by CLASSES2:9;
A19:     X c= rng G by RELAT_1:111; then
A20:     X c= On W by A4;
        reconsider u = union X as Ordinal by A19,A4,ORDINAL3:4,XBOOLE_1:1;
        On W c= W by ORDINAL2:7; then
        X c= W by A20; then
        X in W by A18,CLASSES1:1; then
        u in W by CLASSES2:59; then
        consider b such that
A21:     u in b & b is_a_fixpoint_of F by A1,Th43;
A22:     a c= dom G
        proof
          let c; assume
A23:       c in a; then
          c in W by A17,ORDINAL1:10;
          hence thesis by A16,A23;
        end;
        now
          let x; assume
       x in a; then
          G.x in X by A22,FUNCT_1:def 6; then
          G.x is Ordinal & G.x c= u by ZFMISC_1:74;
          hence G.x in b by A21,ORDINAL1:12;
        end;
        hence thesis by A15,A22,A21,Th38;
      end;
      P[b] from ORDINAL2:sch 1(A8,A9,A14);
      hence thesis by A5;
    end;
    hence thesis by A3,A2,FUNCT_2:2,XBOOLE_1:1;
  end;
