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 Th54:
  for g st dom g <> {} & for a st a in dom g holds g.a is normal
  for a,f st a in dom criticals g & f in rng g holds f.a c= (criticals g).a
  proof let F be Ordinal-Sequence-valued Sequence such that
A1: dom F <> {} and
A2: for a st a in dom F holds F.a is normal;
    let a,f such that
A3: a in dom criticals F and
A4: f in rng F;
    consider x being object such that
A5: x in dom F & f = F.x by A4,FUNCT_1:def 3;
A6: f is normal by A5,A2;
    set g = criticals F;
A7: dom g c= dom f by A4,Th49;
    defpred P[Ordinal] means $1 in dom g implies f.$1 c= g.$1;
A8: P[0]
    proof
      assume 0 in dom g; then
      g.0 is_a_fixpoint_of f by A5,Th47; then
      f.(g.0) = g.0 & g.0 in dom f;
      hence thesis by A6,ORDINAL4:9,XBOOLE_1:2;
    end;
A9: for a holds P[a] implies P[succ a]
    proof let a such that
      P[a] and
A10:   succ a in dom g;
      g.succ a is_a_fixpoint_of f by A5,A10,Th47; then
      g.succ a in dom f & f.(g.succ a) = g.succ a;
      hence f.succ a c= g.succ a by A6,A10,ORDINAL4:9,10;
    end;
A11: for a st a <> 0 & a is limit_ordinal & for b st b in a holds P[b]
    holds P[a]
    proof
      let a such that
A12:   a <> 0 & a is limit_ordinal and
A13:   for b st b in a holds P[b] and
A14:   a in dom g;
      g is continuous by A1,A2,Th53; then
      f.a is_limes_of (f|a) & g.a is_limes_of (g|a)
      by A6,A7,A12,A14,ORDINAL2:def 13; then
A15:   f.a = lim(f|a) & g.a = lim(g|a) by ORDINAL2:def 10;
A16:   f|a is increasing & g|a is increasing by A6,ORDINAL4:15;
A17:   a c= dom f & a c= dom g by A7,A14,ORDINAL1:def 2; then
A18:   dom (f|a) = a & dom (g|a) = a by RELAT_1:62; then
      Union(f|a) is_limes_of (f|a) & Union(g|a) is_limes_of (g|a)
      by A12,A16,ORDINAL5:6; then
A19:   f.a = Union(f|a) & g.a = Union(g|a) by A15,ORDINAL2:def 10;
      let b; assume b in f.a; then
      consider x being object such that
A20:   x in a & b in (f|a).x by A18,A19,CARD_5:2;
      (f|a).x = f.x & (g|a).x = g.x & f.x c= g.x by A17,A13,A20,FUNCT_1:49;
      hence b in g.a by A20,A18,A19,CARD_5:2;
    end;
    for a holds P[a] from ORDINAL2:sch 1(A8,A9,A11);
    hence thesis by A3;
  end;
