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 Th30:
  rng criticals f = {a where a is Element of dom f: a is_a_fixpoint_of f} &
  rng criticals f c= rng f
  proof
    set X = {a where a is Element of dom f: a is_a_fixpoint_of f};
    On X = X by Th28;
    hence
A1: rng criticals f = X by Th18;
    let x be object; assume x in rng criticals f; then
    ex a being Element of dom f st x = a & a is_a_fixpoint_of f by A1;
    hence thesis by Th27;
  end;
