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 Th29:
  x in dom criticals f implies (criticals f).x is_a_fixpoint_of f
  proof set a = x;
    set X = {b where b is Element of dom f: b is_a_fixpoint_of f};
    set g = criticals f;
    assume a in dom g; then
    g.a in rng g by FUNCT_1:def 3; then
    g.a in On X by Th18; then
    g.a in X by Th28; then
    ex b being Element of dom f st g.a = b & b is_a_fixpoint_of f;
    hence thesis;
  end;
