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 Th40:
  for f1,f2 being Ordinal-Sequence st f1 c= f2
  holds criticals f1 c= criticals f2
  proof
    let f1,f2 be Ordinal-Sequence;
    assume
A1: f1 c= f2; then
A2: dom f1 c= dom f2 by GRFUNC_1:2;
    set X = {a where a is Element of dom f1: a is_a_fixpoint_of f1};
    set Z = {a where a is Element of dom f2: a is_a_fixpoint_of f2};
    On X = X & On Z = Z by Th28; then
    reconsider X,Z as ordinal-membered set;
    set Y = Z\X;
A3: now let x,y; assume x in X; then
      consider a being Element of dom f1 such that
A4:   x = a & a is_a_fixpoint_of f1;
      assume y in Y; then
A5:   y in Z & not y in X by XBOOLE_0:def 5; then
      consider b being Element of dom f2 such that
A6:   y = b & b is_a_fixpoint_of f2;
      now assume
A7:     b in dom f1; then
        f1.b = f2.b by A1,GRFUNC_1:2 .= b by A6; then
        b is_a_fixpoint_of f1 by A7;
        hence contradiction by A5,A6;
      end; then
      dom f1 c= b & x in dom f1 by A4,Th4;
      hence x in y by A6;
    end;
    X c= Z
    proof
      let x be object; assume x in X; then
      consider a being Element of dom f1 such that
A8:   x = a & a is_a_fixpoint_of f1;
   a in dom f1 & a = f1.a by A8; then
      a in dom f2 & a = f2.a by A1,A2,GRFUNC_1:2; then
      a is_a_fixpoint_of f2;
      hence thesis by A8;
    end; then
    X\/Y = Z by XBOOLE_1:45; then
    criticals f2 = (criticals f1)^numbering Y by A3,Th25;
    hence criticals f1 c= criticals f2 by Th9;
  end;
