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 Th32:
  dom criticals f c= dom f
  proof let x be Ordinal;
    set X = {a where a is Element of dom f: a is_a_fixpoint_of f};
    assume
A1: x in dom criticals f; then
    (criticals f).x in rng criticals f by FUNCT_1:def 3; then
    (criticals f).x in On X by Th18; then
    (criticals f).x in X by Th28; then
    consider a being Element of dom f such that
A2: (criticals f).x = a & a is_a_fixpoint_of f;
    x c= a & a in dom f & a = f.a by A1,A2,ORDINAL4:10;
    hence thesis by ORDINAL1:12;
  end;
