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
  l in U & (for c st c in l holds a is_a_fixpoint_of U-Veblen.c)
  implies a is_a_fixpoint_of lims(U-Veblen|l)
  proof
    assume
A1: l in U;
    assume
A2: for c st c in l holds a is_a_fixpoint_of U-Veblen.c;
    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;
A3: 0 in l by ORDINAL3:8; then
    omega c= l by ORDINAL1:def 11; then
    reconsider o = omega as non trivial Ordinal of U by A1,CLASSES1:def 1;
    F.0 = U exp o by Def15; then
    reconsider f0 = F.0 as normal Ordinal-Sequence of U;
A4: f0 = g.0 by FUNCT_1:49,ORDINAL3:8; then
A5: dom lims g = dom f0 & dom f0 = On U by Def12,FUNCT_2:def 1;
A6: a is_a_fixpoint_of f0 by A2,ORDINAL3:8; then
A7: a in On U & a = f0.a by A5;
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;
    set lg = lims g;
    thus a in dom lims g by A5,A6;
A10: lg.a = u by A5,A7,Def12;
    {a} = X
    proof
      a in X by A3,A9,A7,A4;
      hence {a} c= X by ZFMISC_1:31;
      let x be object; assume x in X; then
      consider d being Element of dom g such that
A11:   x = g.d.a & d in dom g;
      g.d = F.d by A11,FUNCT_1:47; then
      a is_a_fixpoint_of g.d by A2,A9; then
      x = a by A11;
      hence thesis by TARSKI:def 1;
    end;
    hence a = (lims g).a by A10,ZFMISC_1:25;
  end;
