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 Th56:
  for g being Ordinal-Sequence-valued Sequence
  st dom g <> {} & dom g in U &
  for a st a in dom g holds g.a is Ordinal-Sequence of U
  holds lims g is Ordinal-Sequence of U
  proof
    let g be Ordinal-Sequence-valued Sequence;
    assume
A1: dom g <> {} & dom g in U;
    assume
A2: for a st a in dom g holds g.a is Ordinal-Sequence of U;
    reconsider g0 = g.0 as Ordinal-Sequence of U by A2,A1,ORDINAL3:8;
A3: dom lims g = dom g0 by Def12 .= On U by FUNCT_2:def 1;
    rng lims g c= On U
    proof
      let x be object; assume x in rng lims g; then
      consider y being object such that
A4:   y in dom lims g & x = (lims g).y by FUNCT_1:def 3;
      reconsider y as Ordinal by A4;
      set X = {g.b.y where b is Element of dom g: b in dom g};
A5:   x = union X by A4,Def12;
      reconsider a = dom g as non empty Ordinal of U by A1;
      deffunc F(set) = g.$1.y;
A6:   card {F(b) where b is Element of a: b in a} c= card a from TREES_2:sch 2;
      card a c= a by CARD_1:8; then
      card X c= a by A6; then
      card X in U by CLASSES1:def 1; then
      card X in On U by ORDINAL1:def 9; then
A7:   card X in card U by CLASSES2:9;
A8:   X c= On U
      proof
        let z be object; assume z in X; then
        consider b being Element of a such that
A9:     z = g.b.y & b in a;
        reconsider f = g.b as Ordinal-Sequence of U by A2;
        z = f.y by A9;
        hence thesis by ORDINAL1:def 9;
      end; then
      reconsider u = union X as Ordinal by ORDINAL3:4;
      On U c= U by ORDINAL2:7; then
      X c= U by A8; then
      X in U by A7,CLASSES1:1; then
      u in U by CLASSES2:59;
      hence thesis by A5,ORDINAL1:def 9;
    end;
    hence lims g is Ordinal-Sequence of U by A3,FUNCT_2:2;
  end;
