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;

theorem Th39:
  f is normal & l in dom criticals f implies
  (criticals f).l = Union ((criticals f)|l)
  proof set g = criticals f;
    reconsider h = g|l as increasing Ordinal-Sequence by ORDINAL4:15;
    set X = rng h;
    assume
A1: f is normal & l in dom g; then
    g.l is_a_fixpoint_of f by Th29; then
A2: g.l in dom f & f.(g.l) = g.l;
A3: l c= dom g by A1,ORDINAL1:def 2; then
A4: dom h = l by RELAT_1:62;
A5: for x st x in X holds x is_a_fixpoint_of f
    proof
      let x; assume x in X; then
      consider y being object such that
A6:   y in dom h & x = h.y by FUNCT_1:def 3;
      x = g.y & y in dom g by A3,A4,A6,FUNCT_1:47;
      hence thesis by Th29;
    end;
    reconsider u = union X as Ordinal;
A7: h <> {} by A4;
    now
      let x; assume 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 A3,A4,A8,FUNCT_1:47; then
      x in g.l by A1,A4,A8,ORDINAL2:def 12;
      hence x c= g.l by ORDINAL1:def 2;
    end; then
A9: union X c= g.l by ZFMISC_1:76; then
A10: u in dom f by A2,ORDINAL1:12;
    u = f.u by A1,A5,A7,A9,Th37,A2,ORDINAL1:12; then
    u is_a_fixpoint_of f by A10; then
    consider a such that
A11: a in dom g & u = g.a by Th33;
    a = l
    proof
      thus a c= l by A1,A11,A9,Th22;
      let x be Ordinal; assume
A12:   x in l; then
A13:   succ x in l by ORDINAL1:28; then
A14:   g.x = h.x & g.succ x = h.succ x & h.succ x in X
      by A4,A12,FUNCT_1:47,def 3;
      x in succ x by ORDINAL1:6; then
      h.x in h.succ x by A4,A13,ORDINAL2:def 12; then
      g.x in u by A14,TARSKI:def 4; then
      g.a c/= g.x & x in dom g by A3,A11,A12,Th4; then
      a c/= x by A11,Th22;
      hence thesis by Th4;
    end;
    hence thesis by A11;
  end;
