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;

theorem Th45:
  f is normal implies
  for a st a in dom criticals f holds f.a c= (criticals f).a
  proof assume
A1: f is normal;
    set g = criticals f;
A2: dom g c= dom f by Th32;
    defpred P[Ordinal] means $1 in dom g implies f.$1 c= g.$1;
A3: P[0]
    proof
      assume 0 in dom g; then
      g.0 is_a_fixpoint_of f by Th29; then
      f.(g.0) = g.0 & g.0 in dom f;
      hence thesis by A1,ORDINAL4:9,XBOOLE_1:2;
    end;
A4: P[a] implies P[succ a]
    proof assume that
      P[a] and
A5:   succ a in dom g;
      g.succ a is_a_fixpoint_of f by A5,Th29; then
      g.succ a in dom f & f.(g.succ a) = g.succ a;
      hence f.succ a c= g.succ a by A1,A5,ORDINAL4:9,10;
    end;
A6: for a st a <> 0 & a is limit_ordinal & for b st b in a holds P[b]
    holds P[a]
    proof
      let a such that
A7:   a <> 0 & a is limit_ordinal and
A8:   for b st b in a holds P[b] and
A9:   a in dom g;
      f.a is_limes_of (f|a) & g.a is_limes_of (g|a)
      by A1,A2,A7,A9,ORDINAL2:def 13; then
A10:   f.a = lim(f|a) & g.a = lim(g|a) by ORDINAL2:def 10;
A11:   f|a is increasing & g|a is increasing by A1,ORDINAL4:15;
A12:   a c= dom f & a c= dom g by A2,A9,ORDINAL1:def 2; then
A13:   dom (f|a) = a & dom (g|a) = a by RELAT_1:62; then
      Union(f|a) is_limes_of (f|a) & Union(g|a) is_limes_of (g|a)
      by A7,A11,ORDINAL5:6; then
A14:   f.a = Union(f|a) & g.a = Union(g|a) by A10,ORDINAL2:def 10;
      let b; assume b in f.a; then
      consider x being object such that
A15:   x in a & b in (f|a).x by A13,A14,CARD_5:2;
      (f|a).x = f.x & (g|a).x = g.x & f.x c= g.x by A12,A8,A15,FUNCT_1:49;
      hence b in g.a by A15,A13,A14,CARD_5:2;
    end;
    thus P[a] from ORDINAL2:sch 1(A3,A4,A6);
  end;
