reserve H,S for ZF-formula,
  x for Variable,
  X,Y for set,
  i for Element of NAT,
  e,u for set;
reserve M,M1,M2 for non empty set,
  f for Function,
  v1 for Function of VAR,M1,
  v2 for Function of VAR,M2,
  F,F1,F2 for Subset of WFF,
  W for Universe,
  a,b,c for Ordinal of W,
  A,B,C for Ordinal,
  L for DOMAIN-Sequence of W,
  va for Function of VAR,L.a,
  phi,xi for Ordinal-Sequence of W;
reserve psi for Ordinal-Sequence;

theorem Th28:
  omega in W & phi is increasing & phi is continuous implies ex b
  st a in b & phi.b = b
proof
  assume that
A1: omega in W and
A2: phi is increasing and
A3: phi is continuous;
  deffunc F(Ordinal of W) = (succ a)+^phi.$1;
  consider xi such that
A4: xi.b = F(b) from ORDINAL4:sch 2;
A5: dom xi = On W by FUNCT_2:def 1;
A6: dom phi = On W by FUNCT_2:def 1;
  for A st A in dom phi holds xi.A = (succ a)+^(phi.A)
  proof
    let A;
    assume A in dom phi;
    then reconsider b = A as Ordinal of W by A6,ZF_REFLE:7;
    xi.b = (succ a)+^phi.b by A4;
    hence thesis;
  end;
  then xi = (succ a)+^phi by A6,A5,ORDINAL3:def 1;
  then xi is increasing & xi is continuous by A2,A3,Th14,Th16;
  then consider A such that
A7: A in dom xi and
A8: xi.A = A by A1,ORDINAL4:36;
  reconsider b = A as Ordinal of W by A5,A7,ZF_REFLE:7;
A9: b = (succ a)+^phi.b by A4,A8;
  take b;
A10: succ a c= (succ a)+^phi.b & a in succ a by ORDINAL1:6,ORDINAL3:24;
A11: phi.b c= (succ a)+^phi.b by ORDINAL3:24;
  b c= phi.b by A2,A6,A5,A7,ORDINAL4:10;
  hence thesis by A10,A9,A11;
end;
