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 Th49:
  f in rng g implies dom criticals g c= dom f
  proof assume
A1: f in rng g;
    let x be Ordinal;
    set X = {a where a is Element of dom(g.0): a in dom(g.0) &
    for f st f in rng g holds a is_a_fixpoint_of f};
    assume
A2: x in dom criticals g; then
    (criticals g).x in rng criticals g by FUNCT_1:def 3; then
    (criticals g).x in On X & X is ordinal-membered by Th18,Th46; then
    (criticals g).x in X by Th2; then
    consider a being Element of dom(g.0) such that
A3: (criticals g).x = a & a in dom(g.0) &
    for f st f in rng g holds a is_a_fixpoint_of f;
    a is_a_fixpoint_of f by A1,A3; then
    x c= a & a in dom f & a = f.a by A2,A3,ORDINAL4:10;
    hence thesis by ORDINAL1:12;
  end;
