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 Th52:
  for g st dom g <> {} & for a st a in dom g holds g.a is normal
  holds l in dom criticals g implies
  (criticals g).l = Union ((criticals g)|l)
  proof let F be Ordinal-Sequence-valued Sequence such that
A1: dom F <> {};
    set g = criticals F;
    reconsider h = g|l as increasing Ordinal-Sequence by ORDINAL4:15;
    set X = rng h;
    assume
A2: (for a st a in dom F holds F.a is normal) & l in dom g;
A3: now
      let a; set f = F.a;
      assume a in dom F; then
      g.l is_a_fixpoint_of F.a by A2,Th47;
      hence g.l in dom f & f.(g.l) = g.l;
    end;
A4: l c= dom g by A2,ORDINAL1:def 2; then
A5: dom h = l by RELAT_1:62;
A6: for a,x st a in dom F & x in X holds x is_a_fixpoint_of F.a
    proof
      let a,x; assume
A7:   a in dom F & x in X; then
      consider y being object such that
A8:   y in dom h & x = h.y by FUNCT_1:def 3;
      x = g.y & y in dom g by A4,A5,A8,FUNCT_1:47;
      hence thesis by A7,Th47;
    end;
    reconsider u = union X as Ordinal;
A9: h <> {} by A5;
    now
      let x; assume x in X; then
      consider y being object such that
A10:   y in dom h & x = h.y by FUNCT_1:def 3;
      x = g.y & y in dom g by A4,A5,A10,FUNCT_1:47; then
      x in g.l by A2,A5,A10,ORDINAL2:def 12;
      hence x c= g.l by ORDINAL1:def 2;
    end; then
A11: u c= g.l by ZFMISC_1:76;
    now let c; set f = F.c;
      assume
A12:   c in dom F; then
A13:   g.l in dom f by A3; then
A14:   u in dom f by A11,ORDINAL1:12;
A15:   f is normal by A2,A12;
      for x st x in X holds x is_a_fixpoint_of f by A6,A12; then
      u = f.u by A9,A13,A15,Th37,A11,ORDINAL1:12;
      hence u is_a_fixpoint_of f by A14;
    end; then
    consider a such that
A16: a in dom g & u = g.a by A1,Th50;
    a = l
    proof
      thus a c= l by A2,A16,A11,Th22;
      let x be Ordinal; assume
A17:   x in l; then
A18:   succ x in l by ORDINAL1:28; then
A19:   g.x = h.x & g.succ x = h.succ x & h.succ x in X
      by A5,A17,FUNCT_1:47,def 3;
      x in succ x by ORDINAL1:6; then
      h.x in h.succ x by A5,A18,ORDINAL2:def 12; then
      g.x in u by A19,TARSKI:def 4; then
      g.a c/= g.x & x in dom g by A4,A16,A17,Th4; then
      a c/= x by A16,Th22;
      hence thesis by Th4;
    end;
    hence thesis by A16;
  end;
