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 a st a in l holds
  U-Veblen.a is normal Ordinal-Sequence of U)
  implies lims(U-Veblen|l) is non-decreasing continuous Ordinal-Sequence of U
  proof set G = U-Veblen;
    assume that
A1: l in U and
A2: for a st a in l holds G.a is normal Ordinal-Sequence of U;
    0 in l by ORDINAL3:8; then
    omega c= l by ORDINAL1:def 11; then
A3: omega in U & l in On U by A1,CLASSES1:def 1,ORDINAL1:def 9; then
A4: G.l = criticals(G|l) & dom G = On U by Def15;
    l c= On U by A3,ORDINAL1:def 2; then
A5: dom(G|l) = l by A4,RELAT_1:62;
    set g = G|l;
    now let a; assume
A6:   a in dom g; then
      g.a = G.a by A5,FUNCT_1:49;
      hence g.a is Ordinal-Sequence of U by A2,A5,A6;
    end; then
    reconsider f = lims g as Ordinal-Sequence of U by A1,A5,Th56;
    g.0 = G.0 by FUNCT_1:49,ORDINAL3:8; then
    reconsider g0 = g.0 as Ordinal-Sequence of U by A2,ORDINAL3:8;
A7: dom f = On U by FUNCT_2:def 1;
    deffunc X(object) = {g.b.$1 where b is Element of dom g: b in dom g};
A8: f is non-decreasing
    proof
      let a,b; assume
A9:   a in b & b in dom f; then
      a in dom f by ORDINAL1:10; then
A10:   f.a = union X(a) & f.b = union X(b) by A9,Def12;
      let c; assume c in f.a; then
      consider x such that
A11:   c in x & x in X(a) by A10,TARSKI:def 4;
      consider xa being Element of dom g such that
A12:   x = g.xa.a & xa in dom g by A11;
      g.xa = G.xa by A5,FUNCT_1:49; then
      reconsider h = g.xa as increasing Ordinal-Sequence of U by A2,A5;
      dom h = On U by FUNCT_2:def 1; then
      h.a in h.b by A7,A9,ORDINAL2:def 12; then
      h.a c= h.b by ORDINAL1:def 2; then
      c in h.b & h.b in X(b) by A11,A12;
      hence c in f.b by A10,TARSKI:def 4;
    end;
    f is continuous
    proof let a,b; assume
A13:   a in dom f & a <> 0 & a is limit_ordinal & b = f.a; then
A14:   b = union X(a) by Def12;
A15:   a c= dom f by A13,ORDINAL1:def 2; then
A16:   dom(f|a) = a by RELAT_1:62;
A17:   b = Union(f|a)
      proof
        thus b c= Union(f|a)
        proof
          let c; assume c in b; then
          consider x such that
A18:       c in x & x in X(a) by A14,TARSKI:def 4;
          consider xa being Element of dom g such that
A19:       x = g.xa.a & xa in dom g by A18;
          g.xa = G.xa by A5,FUNCT_1:49; then
          reconsider h = g.xa as normal Ordinal-Sequence of U
          by A2,A5;
A20:       dom h = On U by FUNCT_2:def 1; then
A21:       h.a is_limes_of h|a by A7,A13,ORDINAL2:def 13;
A22:       h|a is increasing by ORDINAL4:15;
A23:       dom(h|a) = a by A7,A15,A20,RELAT_1:62; then
          Union(h|a) is_limes_of h|a by A13,A22,ORDINAL5:6; then
          lim(h|a) = Union(h|a) by ORDINAL2:def 10; then
          h.a = Union(h|a) by A21,ORDINAL2:def 10; then
          consider y being object such that
A24:       y in a & c in (h|a).y by A18,A19,A23,CARD_5:2;
A25:       y in On U by A7,A13,A24,ORDINAL1:10;
          (h|a).y = h.y by A24,FUNCT_1:49; then
          (h|a).y in X(y) by A19; then
          c in union X(y) by A24,TARSKI:def 4; then
          c in f.y by A7,A25,Def12; then
          c in (f|a).y by A24,FUNCT_1:49;
          hence thesis by A16,A24,CARD_5:2;
        end;
        let c; assume c in Union(f|a); then
        consider x being object such that
A26:     x in dom(f|a) & c in (f|a).x by CARD_5:2;
A27:     (f|a).x = f.x by A16,A26,FUNCT_1:49;
        x in dom f by A26,RELAT_1:57; then
        f.x = union X(x) by Def12; then
        consider y such that
A28:     c in y & y in X(x) by A26,A27,TARSKI:def 4;
        consider z being Element of dom g such that
A29:     y = g.z.x & z in dom g by A28;
        g.z = G.z by A5,FUNCT_1:49; then
        reconsider h = g.z as normal Ordinal-Sequence of U
        by A2,A5;
        dom h = On U by FUNCT_2:def 1; then
        h.x in h.a by A26,A16,A13,A7,ORDINAL2:def 12; then
        h.x c= h.a by ORDINAL1:def 2; then
        c in h.a & h.a in X(a) by A28,A29;
        hence c in b by A14,TARSKI:def 4;
      end;
      f|a is non-decreasing by A8,Th13;
      hence b is_limes_of f|a by A13,A16,A17,ORDINAL5:6;
    end;
    hence thesis by A8;
  end;
