reserve phi,fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  f,g for Function,
  X for set,
  x,y,z for object;
reserve f1,f2 for Ordinal-Sequence;
reserve W for Universe;
reserve A1,B1 for Ordinal of W,
  phi for Ordinal-Sequence of W;
reserve L for Sequence;

theorem
  phi is increasing & phi is continuous & omega in W implies ex A st A
  in dom phi & phi.A = A
proof
  deffunc D(set,set) = {};
  assume that
A1: phi is increasing and
A2: phi is continuous and
A3: omega in W;
  deffunc C(set,set) = phi.$2;
  reconsider N = phi.(0-element_of W) as Ordinal;
  consider L such that
A4: dom L = omega and
A5: 0 in omega implies L.0 = N and
A6: for A st succ A in omega holds L.(succ A) = C(A,L.A) and
  for A st A in omega & A <> 0 & A is limit_ordinal holds L.A = D(A,L|A)
  from ORDINAL2:sch 5;
  defpred P[Ordinal] means $1 in dom L implies L.$1 is Ordinal of W;
A7: for A st P[A] holds P[succ A]
  proof
    let A such that
A8: A in dom L implies L.A is Ordinal of W and
A9: succ A in dom L;
    A in succ A by ORDINAL1:6;
    then reconsider x = L.A as Ordinal of W by A8,A9,ORDINAL1:10;
    L.succ A = phi.x by A4,A6,A9;
    hence thesis;
  end;
A10: 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
A11: A <> 0 and
A12: A is limit_ordinal and
    for B st B in A holds B in dom L implies L.B is Ordinal of W and
A13: A in dom L;
    {} in A by A11,ORDINAL3:8;
    then omega c= A by A12,ORDINAL1:def 11;
    hence thesis by A4,A13,ORDINAL1:5;
  end;
A14: P[0] by A4,A5;
A15: for A holds P[A] from ORDINAL2:sch 1(A14,A7,A10);
  rng L c= sup rng L
  proof
    let x be object;
    assume
A16: x in rng L;
    then consider y being object such that
A17: y in dom L and
A18: x = L.y by FUNCT_1:def 3;
    reconsider y as Ordinal by A17;
    reconsider A = L.y as Ordinal of W by A15,A17;
    A in sup rng L by A16,A18,ORDINAL2:19;
    hence thesis by A18;
  end;
  then reconsider L as Ordinal-Sequence by ORDINAL2:def 4;
A19: dom phi = On W by FUNCT_2:def 1;
  assume
A20: not thesis;
A21: now
    let A1;
    A1 in dom phi by Th34;
    then
A22: A1 c= phi.A1 by A1,Th10;
    A1 <> phi.A1 by A20,Th34;
    then A1 c< phi.A1 by A22;
    hence A1 in phi.A1 by ORDINAL1:11;
  end;
  L is increasing
  proof
    let A,B;
    assume that
A23: A in B and
A24: B in dom L;
    defpred P[Ordinal] means A+^$1 in omega & $1 <> {} implies L.A in L.(A+^$1
    );
A25: for B st B <> 0 & B is limit_ordinal & for C st C in B holds P[C]
    holds P[B]
    proof
      let B such that
A26:  B <> 0 and
A27:  B is limit_ordinal and
      for C st C in B holds A+^C in omega & C <> {} implies L.A in L.(A+^C
      ) and
A28:  A+^B in omega and
A29:  B <> {};
      A+^B <> {} by A29,ORDINAL3:26;
      then
A30:  {} in A+^B by ORDINAL3:8;
      A+^B is limit_ordinal by A26,A27,ORDINAL3:29;
      then omega c= A+^B by A30,ORDINAL1:def 11;
      hence thesis by A28,ORDINAL1:5;
    end;
A31: for C st P[C] holds P[succ C]
    proof
      let C such that
A32:  A+^C in omega & C <> {} implies L.A in L.(A+^C) and
A33:  A+^succ C in omega and
      succ C <> {};
A34:  A+^succ C = succ(A+^C) by ORDINAL2:28;
A35:  A+^C in succ(A+^C) by ORDINAL1:6;
      then reconsider D = L.(A+^C) as Ordinal of W by A4,A15,A33,A34,
ORDINAL1:10;
A36:  D in phi.D by A21;
      L.(A+^succ C) = phi.D by A6,A33,A34;
      hence thesis by A32,A33,A35,A34,A36,ORDINAL1:10,ORDINAL2:27;
    end;
A37: P[0];
A38: for C holds P[C] from ORDINAL2:sch 1(A37,A31,A25);
    ex C st B = A+^C & C <> {} by A23,ORDINAL3:28;
    hence thesis by A4,A24,A38;
  end;
  then
A39: sup L is limit_ordinal by A4,Lm2,Th16;
A40: rng L c= W
  proof
    let x be object;
    assume x in rng L;
    then consider y being object such that
A41: y in dom L and
A42: x = L.y by FUNCT_1:def 3;
    reconsider y as Ordinal by A41;
    L.y is Ordinal of W by A15,A41;
    hence thesis by A42;
  end;
  then reconsider S = sup L as Ordinal of W by A3,A4,Th35;
  set fi = phi|sup L;
  N in rng L by A4,A5,Lm1,FUNCT_1:def 3;
  then
A43: sup L <> {} by ORDINAL2:19;
A44: S in On W by ORDINAL1:def 9;
  then
A45: phi.S is_limes_of fi by A2,A43,A39,A19;
  S c= dom phi by A44,A19,ORDINAL1:def 2;
  then
A46: dom fi = S by RELAT_1:62;
A47: sup fi c= sup L
  proof
    let x be object;
    assume
A48: x in sup fi;
    then reconsider A = x as Ordinal;
    consider B such that
A49: B in rng fi and
A50: A c= B by A48,ORDINAL2:21;
    consider y being object such that
A51: y in dom fi and
A52: B = fi.y by A49,FUNCT_1:def 3;
    reconsider y as Ordinal by A51;
    consider C such that
A53: C in rng L and
A54: y c= C by A46,A51,ORDINAL2:21;
    reconsider C1 = C as Ordinal of W by A40,A53;
    y c< C1 iff y c= C1 & y <> C1;
    then y in C1 & C1 in dom phi or y = C by A19,A54,ORDINAL1:11,def 9;
    then
A55: phi.y in phi.C1 or y = C1 by A1;
    B = phi.y by A51,A52,FUNCT_1:47;
    then
A56: B c= phi.C1 by A55,ORDINAL1:def 2;
    consider z being object such that
A57: z in dom L and
A58: C = L.z by A53,FUNCT_1:def 3;
    reconsider z as Ordinal by A57;
A59: succ z in omega by A4,A57,Lm2,ORDINAL1:28;
    then
A60: L.succ z in rng L by A4,FUNCT_1:def 3;
    L.succ z = phi.C by A6,A58,A59;
    then phi.C1 in sup L by A60,ORDINAL2:19;
    then B in sup L by A56,ORDINAL1:12;
    hence thesis by A50,ORDINAL1:12;
  end;
  fi is increasing
  proof
A61: dom fi c= dom phi by RELAT_1:60;
    let A,B;
    assume that
A62: A in B and
A63: B in dom fi;
A64: fi.B = phi.B by A63,FUNCT_1:47;
    fi.A = phi.A by A62,A63,FUNCT_1:47,ORDINAL1:10;
    hence thesis by A1,A62,A63,A61,A64;
  end;
  then sup fi = lim fi by A43,A39,A46,Th8
    .= phi.(sup L) by A45,ORDINAL2:def 10;
  then not S in phi.S by A47,ORDINAL1:5;
  hence contradiction by A21;
end;
