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;
reserve g for Ordinal-Sequence-valued Sequence;

theorem Th58:
  a in b & b in U & omega in U & c in dom(U-Veblen.b)
  implies U-Veblen.b.c is_a_fixpoint_of U-Veblen.a
  proof assume
A1: a in b & b in U & omega in U;
    set F = U-Veblen;
    defpred P[Ordinal] means $1 in U implies
    for a,c st a in $1 & c in dom(F.$1) holds F.$1.c is_a_fixpoint_of F.a;
A2: P[0];
A3: for b st P[b] holds P[succ b]
    proof
      let b such that
A4:   P[b] and
A5:   succ b in U;
A6:   b in succ b by ORDINAL1:6;
      let a,c;
      assume a in succ b; then
A7:   a in b or a in {b} by XBOOLE_0:def 3;
      succ b in On U by A5,ORDINAL1:def 9; then
A8:   F.succ b = criticals(F.b) by Def15;
      assume c in dom(F.succ b); then
   F.(succ b).c is_a_fixpoint_of F.b by A8,Th29; then
      F.(succ b).c in dom(F.b) & F.(succ b).c = F.b.(F.(succ b).c);
      hence thesis by A4,A5,A6,A7,ORDINAL1:10,TARSKI:def 1;
    end;
A9: dom F = On U by Def15;
A10: for b st b <> 0 & b is limit_ordinal & for d st d in b holds P[d]
    holds P[b]
    proof
      let b such that
A11:   b <> 0 & b is limit_ordinal and
      for d st d in b holds P[d] and
A12:   b in U;
A13:   b in On U by A12,ORDINAL1:def 9; then
A14:   F.b = criticals(F|b) by A11,Def15;
      b c= On U by A13,ORDINAL1:def 2; then
A15:   dom(F|b) = b by A9,RELAT_1:62;
      let a,c; assume
A16:   a in b; then
A17:   F.a = (F|b).a by FUNCT_1:49;
      set g = F|b;
      set X = {z where z is Element of dom(g.0): z in dom(g.0) &
      for f st f in rng g holds z is_a_fixpoint_of f};
      now
        let x; assume x in X; then
        ex a being Element of dom(g.0) st x = a & a in dom(g.0) &
        for f st f in rng g holds a is_a_fixpoint_of f;
        hence x is ordinal;
      end; then
      reconsider X as ordinal-membered set by Th1;
      assume
      c in dom(F.b); then
      F.b.c in rng (F.b) by FUNCT_1:def 3; then
      F.b.c in X by A14,Th19; then
      consider z being Element of dom(g.0) such that
A18:   F.b.c = z & z in dom(g.0) &
      for f st f in rng g holds z is_a_fixpoint_of f;
      F.a in rng g by A15,A17,A16,FUNCT_1:def 3;
      hence thesis by A18;
    end;
    for b holds P[b] from ORDINAL2:sch 1(A2,A3,A10);
    hence thesis by A1;
  end;
