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 Th55:
  for g1,g2 being Ordinal-Sequence-valued Sequence
  st dom g1 = dom g2 & dom g1 <> {} & for a st a in dom g1 holds g1.a c= g2.a
  holds criticals g1 c= criticals g2
  proof
    let g1,g2 be Ordinal-Sequence-valued Sequence;
    assume
A1: dom g1 = dom g2;
    assume
A2: dom g1 <> {};
    assume
A3: for a st a in dom g1 holds g1.a c= g2.a;
    set f1 = g1.0, f2 = g2.0;
    0 in dom g1 by A2,ORDINAL3:8; then
    f1 c= f2 & f1 in rng g1 & f2 in rng g2 by A1,A3,FUNCT_1:def 3; then
A4: dom f1 c= dom f2 by GRFUNC_1:2;
    set X = {a where a is Element of dom f1: a in dom f1 &
    for f st f in rng g1 holds a is_a_fixpoint_of f};
    set Z = {a where a is Element of dom f2: a in dom f2 &
    for f st f in rng g2 holds a is_a_fixpoint_of f};
    reconsider X,Z as ordinal-membered set by Th46;
    set Y = Z\X;
A5: now let x,y; assume x in X; then
      consider a being Element of dom f1 such that
A6:   x = a & a in dom f1 & for f st f in rng g1 holds a is_a_fixpoint_of f;
      assume y in Y; then
A7:   y in Z & not y in X by XBOOLE_0:def 5; then
      consider b being Element of dom f2 such that
A8:   y = b & b in dom f2 & for f st f in rng g2 holds b is_a_fixpoint_of f;
      assume not x in y; then
A9:   b c= a by A6,A8,ORDINAL1:16; then
A10:   b in dom f1 by A6,ORDINAL1:12;
      now
        let f; assume
A11:     f in rng g1; then
        consider z being object such that
A12:     z in dom g1 & f = g1.z by FUNCT_1:def 3;
        reconsider z as set by TARSKI:1;
A13:     f c= g2.z by A3,A12;
        g2.z in rng g2 by A1,A12,FUNCT_1:def 3; then
A14:     b is_a_fixpoint_of g2.z by A8;
        a is_a_fixpoint_of f by A6,A11; then
        a in dom f; then
A15:     b in dom f by A9,ORDINAL1:12; then
        f.b = g2.z.b by A13,GRFUNC_1:2 .= b by A14;
        hence b is_a_fixpoint_of f by A15;
      end;
      hence contradiction by A7,A8,A10;
    end;
    X c= Z
    proof
      let x be object; assume x in X; then
      consider a being Element of dom f1 such that
A16:   x = a & a in dom f1 & for f st f in rng g1 holds a is_a_fixpoint_of f;
      now let f; assume
        f in rng g2; then
        consider z being object such that
A17:     z in dom g2 & f = g2.z by FUNCT_1:def 3;
       reconsider z as set by TARSKI:1;
A18:     g1.z c= f by A1,A3,A17; then
A19:     dom(g1.z) c= dom f by GRFUNC_1:2;
        g1.z in rng g1 by A1,A17,FUNCT_1:def 3; then
        a is_a_fixpoint_of g1.z by A16; then
        a in dom(g1.z) & a = g1.z.a; then
        a in dom f & a = f.a by A18,A19,GRFUNC_1:2;
        hence a is_a_fixpoint_of f;
      end;
      hence thesis by A16,A4;
    end; then
    X\/Y = Z by XBOOLE_1:45; then
    criticals g2 = (criticals g1)^numbering Y by A5,Th25;
    hence criticals g1 c= criticals g2 by Th9;
  end;
