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 Th61:
  l in U & a in U & b in l &
  (for c st c in l holds U-Veblen.c is normal Ordinal-Sequence of U)
  implies (lims(U-Veblen|l)).a is_a_fixpoint_of U-Veblen.b
  proof assume that
A1: l in U and
A2: a in U and
A3: b in l and
A4: for c st c in l holds U-Veblen.c is normal Ordinal-Sequence of U;
    set F = U-Veblen;
    set g = F|l;
    set X = {g.d.a where d is Element of dom g: d in dom g};
    set u = union X;
A5: 0 in l by ORDINAL3:8;
    reconsider f0 = F.0, f = F.b as normal Ordinal-Sequence of U
    by A3,A4,ORDINAL3:8;
A6: f0 = g.0 & f = g.b by A3,FUNCT_1:49,ORDINAL3:8; then
A7: dom lims g = dom f0 by Def12 .= On U by FUNCT_2:def 1;
A8: dom F = On U by Def15;
    l in On U by A1,ORDINAL1:def 9; then
    l c= dom F by A8,ORDINAL1:def 2; then
A9: dom g = l by RELAT_1:62; then
A10: g.b.a in X by A3;
    now
      let c; assume
A11:   c in dom g; then
      g.c = F.c by FUNCT_1:47;
      hence g.c is Ordinal-Sequence of U by A9,A11,A4;
    end; then
    reconsider lg = lims g as Ordinal-Sequence of U by A1,A9,Th56;
A12: a in On U by A2,ORDINAL1:def 9; then
A13: lg.a = u by A7,Def12;
A14: dom f = On U & dom f0 = On U by FUNCT_2:def 1;
A15: for x st x in X ex y st x c= y & y in X & y is_a_fixpoint_of f
    proof let x; assume
A16:   x in X; then
      consider d being Element of dom g such that
A17:   x = g.d.a & d in dom g;
      reconsider f2 = F.d as normal Ordinal-Sequence of U by A4,A9;
A18:   f2 = g.d by A9,FUNCT_1:49;
A19:   dom f2 = On U by FUNCT_2:def 1;
      omega c= l by A5,ORDINAL1:def 11; then
A20:   d in U & omega in U by A9,A1,CLASSES1:def 1,ORDINAL1:10;
A21:   b in U by A1,A3,ORDINAL1:10;
A22:   for c st c in b holds U-Veblen.c is normal by A4,A3,ORDINAL1:10;
      per cases by ORDINAL1:16;
      suppose
        d c= b; then
A23:     x c= g.b.a by A12,A6,A14,A17,A18,A20,A21,A22,Th60;
        take y = g.(succ b).a;
A24:     b in succ b & succ b in l by A3,ORDINAL1:6,28; then
        reconsider f1 = F.succ b as normal Ordinal-Sequence of U by A4;
A25:     f1 = g.succ b by A24,FUNCT_1:49;
        succ b in U by A24,A1,ORDINAL1:10; then
        succ b in On U by ORDINAL1:def 9; then
A26:     f1 = criticals f & dom f1 = On U by Def15,FUNCT_2:def 1; then
        f.a c= y by A12,A25,Th45;
        hence x c= y by A23,A6;
        thus y in X by A9,A24;
        thus thesis by A12,A25,A26,Th29;
      end;
      suppose
A27:     b in d;
        take y = x;
        thus x c= y & y in X by A16;
        thus thesis by A12,A17,A27,A18,A19,A20,Th58;
      end;
    end;
    thus (lims(U-Veblen|l)).a in dom(U-Veblen.b) by A13,A14,ORDINAL1:def 9;
    hence thesis by A13,A10,A15,Th36;
  end;
