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 Th38:
  l c= dom criticals f & a is_a_fixpoint_of f &
  (for x st x in l holds (criticals f).x in a)
  implies l in dom criticals f
  proof set g = criticals f;
    assume that
A1: l c= dom g and
A2: a is_a_fixpoint_of f and
A3: for x st x in l holds g.x in a;
    consider b such that
A4: b in dom g & a = g.b by A2,Th33;
    l c= b
    proof
      let x be Ordinal; assume x in l; then
      x in dom g & g.x in g.b by A1,A3,A4;
      hence x in b by A4,Th23;
    end;
    hence l in dom criticals f by A4,ORDINAL1:12;
  end;
