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;

theorem Th13:
  for D being non empty set, Phi being Function of D, Funcs(On W,
On W) st card D in card W ex phi st phi is increasing & phi is continuous & phi
.0-element_of W = 0-element_of W & (for a holds phi.succ a = sup ({phi.a} \/ (
  uncurry Phi).:[:D,{succ a}:])) & for a st a <> 0-element_of W & a is
  limit_ordinal holds phi.a = sup (phi|a)
proof
  deffunc D(set,Sequence) = sup $2;
  let D be non empty set, Phi be Function of D, Funcs(On W, On W) such that
A1: card D in card W;
  deffunc C(Ordinal,Ordinal) = sup ({$2} \/ (uncurry Phi).:[:D,{succ $1}:]);
  consider L be Ordinal-Sequence such that
A2: dom L = On W & (0 in On W implies L.0 = 0) and
A3: for A st succ A in On W holds L.succ A = C(A,L.A) and
A4: for A st A in On W & A <> 0 & A is limit_ordinal holds L.A = D(A,L|
  A) from ORDINAL2:sch 11;
  defpred P[Ordinal] means L.$1 in On W;
A5: for a st P[a] holds P[succ a]
  proof
    let a;
A6: On W c= W by ORDINAL2:7;
    assume L.a in On W;
    then reconsider b = L.a as Ordinal of W by ZF_REFLE:7;
    card [:D,{succ a}:] = card D by CARD_1:69;
    then card ((uncurry Phi).:[:D,{succ a}:]) c= card D by CARD_1:66;
    then
A7: card ((uncurry Phi).:[:D,{succ a}:]) in card W by A1,ORDINAL1:12;
    rng Phi c= Funcs(On W, On W) by RELAT_1:def 19;
    then
A8: rng uncurry Phi c= On W by FUNCT_5:41;
    (uncurry Phi).:[:D,{succ a}:] c= rng uncurry Phi by RELAT_1:111;
    then (uncurry Phi).:[:D,{succ a}:] c= On W by A8;
    then (uncurry Phi).:[:D,{succ a}:] c= W by A6;
    then (uncurry Phi).:[:D,{succ a}:] in W by A7,CLASSES1:1;
    then
A9: {b} \/ (uncurry Phi).:[:D,{succ a}:] in W by CLASSES2:60;
    succ a in On W by ZF_REFLE:7;
    then L.succ a = sup ({b} \/ (uncurry Phi).:[:D,{succ a}:]) by A3;
    then L.succ a in W by A9,ZF_REFLE:19;
    hence thesis by ORDINAL1:def 9;
  end;
A10: for a st a <> 0-element_of W & a is limit_ordinal & for b st b in a
  holds P[b] holds P[a]
  proof
    let a such that
A11: a <> 0-element_of W & a is limit_ordinal and
A12: for b st b in a holds L.b in On W;
A13: dom (L|a) c= a by RELAT_1:58;
    then
A14: dom (L|a) in W by CLASSES1:def 1;
A15: a in On W by ZF_REFLE:7;
A16: rng (L|a) c= W
    proof
      let e be object;
      assume e in rng (L|a);
      then consider u being object such that
A17:  u in dom (L|a) and
A18:  e = (L|a).u by FUNCT_1:def 3;
      reconsider u as Ordinal by A17;
      u in On W by A15,A13,A17,ORDINAL1:10;
      then reconsider u as Ordinal of W by ZF_REFLE:7;
      e = L.u by A17,A18,FUNCT_1:47;
      then e in On W by A12,A13,A17;
      hence thesis by ORDINAL1:def 9;
    end;
    L.a = sup (L|a) by A4,A11,A15;
    then L.a in W by A14,A16,Th11,ZF_REFLE:19;
    hence thesis by ORDINAL1:def 9;
  end;
A19: P[0-element_of W] by A2,ORDINAL1:def 9;
  rng L c= On W
  proof
    let e be object;
    assume e in rng L;
    then consider u being object such that
A20: u in dom L and
A21: e = L.u by FUNCT_1:def 3;
    reconsider u as Ordinal of W by A2,A20,ZF_REFLE:7;
    P[a] from ZF_REFLE:sch 4(A19,A5,A10);
    then L.u in On W;
    hence thesis by A21;
  end;
  then reconsider phi = L as Ordinal-Sequence of W by A2,FUNCT_2:def 1
,RELSET_1:4;
  take phi;
  thus
A22: phi is increasing
  proof
    let A,B;
    assume that
A23: A in B and
A24: B in dom phi;
    A in dom phi by A23,A24,ORDINAL1:10;
    then reconsider a = A, b = B as Ordinal of W by A2,A24,ZF_REFLE:7;
    defpred Q[Ordinal] means a in $1 implies phi.a in phi.$1;
A25: for b st Q[b] holds Q[succ b]
    proof
      let b such that
A26:  ( a in b implies phi.a in phi.b)& a in succ b;
      phi.b in {phi.b} by TARSKI:def 1;
      then
A27:  phi.b in {phi.b} \/ (uncurry Phi).:[:D,{succ b}:] by XBOOLE_0:def 3;
      succ b in On W by ZF_REFLE:7;
      then phi.succ b = sup({phi.b} \/ (uncurry Phi).:[:D,{succ b}:]) by A3;
      then phi.b in phi.succ b by A27,ORDINAL2:19;
      hence thesis by A26,ORDINAL1:8,10;
    end;
A28: for b st b <> 0-element_of W & b is limit_ordinal & for c st c in b
    holds Q[c] holds Q[b]
    proof
      let b such that
A29:  b <> 0-element_of W & b is limit_ordinal and
      for c st c in b holds a in c implies phi.a in phi.c and
A30:  a in b;
      b in On W by ZF_REFLE:7;
      then
A31:  phi.b = sup (phi|b) by A4,A29;
      a in On W by ZF_REFLE:7;
      then phi.a in rng (phi|b) by A2,A30,FUNCT_1:50;
      hence phi.a in phi.b by A31,ORDINAL2:19;
    end;
A32: Q[0-element_of W];
    for b holds Q[b] from ZF_REFLE:sch 4(A32,A25,A28);
    then phi.a in phi.b by A23;
    hence thesis;
  end;
  thus phi is continuous
  proof
    let A,B;
    assume that
A33: A in dom phi and
A34: A <> 0 & A is limit_ordinal and
A35: B = phi.A;
    A c= dom phi by A33,ORDINAL1:def 2;
    then
A36: dom (phi|A) = A by RELAT_1:62;
A37: phi|A is increasing by A22,ORDINAL4:15;
    B = sup (phi|A) by A2,A4,A33,A34,A35;
    hence thesis by A34,A36,A37,ORDINAL4:8;
  end;
  thus phi.0-element_of W = 0-element_of W by A2,ORDINAL1:def 9;
  thus for a holds phi.succ a = sup ({phi.a} \/ (uncurry Phi).:[:D,{succ a}:] )
  by A3,ZF_REFLE:7;
  let a;
  a in On W by ZF_REFLE:7;
  hence thesis by A4;
end;
