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
  dom g <> {} & l c= dom criticals g &
  (for f st f in rng g holds a is_a_fixpoint_of f) &
  (for x st x in l holds (criticals g).x in a)
  implies l in dom criticals g
  proof set h = criticals g;
    assume that
A1: dom g <> {} and
A2: l c= dom h and
A3: for f st f in rng g holds a is_a_fixpoint_of f and
A4: for x st x in l holds h.x in a;
    now
      let c; assume
      c in dom g; then
      g.c in rng g by FUNCT_1:def 3;
      hence a is_a_fixpoint_of g.c by A3;
    end; then
    consider b such that
A5: b in dom h & a = h.b by A1,Th50;
    l c= b
    proof
      let x be Ordinal; assume x in l; then
      x in dom h & h.x in h.b by A2,A4,A5;
      hence x in b by A5,Th23;
    end;
    hence l in dom criticals g by A5,ORDINAL1:12;
  end;
