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 Th43:
  omega in W & b in W implies ex a st b in a & a is_a_fixpoint_of phi
  proof assume that
A1: omega in W and
A2: b in W;
    reconsider b1 = b as Ordinal of W by A2;
A3: dom phi = On W by FUNCT_2:def 1;
    deffunc phi(set) = phi.$1;
A4: for a st a in W holds phi(a) in W;
A5: for a,b st a in b & b in W holds phi(a) in phi(b)
    proof
      let a,b;
      b in W implies b in dom phi by A3,ORDINAL1:def 9;
      hence thesis by ORDINAL2:def 12;
    end;
A6: for a being Ordinal of W st a is non empty limit_ordinal
    for f being Ordinal-Sequence
      st dom f = a & for b st b in a holds f.b = phi(b)
    holds phi(a) is_limes_of f
    proof let a be Ordinal of W;
      assume
A7:   a is non empty limit_ordinal;
      let f be Ordinal-Sequence such that
A8:   dom f = a and
A9:   for b st b in a holds f.b = phi(b);
A10:   a in dom phi by A3,ORDINAL1:def 9; then
      a c= dom phi by ORDINAL1:def 2; then
A11:   dom (phi|a) = a by RELAT_1:62;
      now let x be object; assume
A12:     x in a;
         reconsider xx = x as set by TARSKI:1;
        thus (phi|a).x = phi(xx) by A12,FUNCT_1:49 .= f.x by A12,A9;
      end; then
      f = phi|a by A8,A11;
      hence phi(a) is_limes_of f by A7,A10,ORDINAL2:def 13;
    end;
    consider a being Ordinal of W such that
A13: b1 in a & phi(a) = a from CriticalNumber3(A1,A4,A5,A6);
    take a;
    thus b in a & a in dom phi & a = phi.a by A3,A13,ORDINAL1:def 9;
  end;
