reserve            x for object,
               X,Y,Z for set,
         i,j,k,l,m,n for Nat,
                 r,s for Real,
                  no for Element of OrderedNAT,
                   A for Subset of [:NAT,NAT:];
reserve X,Y,X1,X2 for non empty set,
          cA1,cB1 for filter_base of X1,
          cA2,cB2 for filter_base of X2,
              cF1 for Filter of X1,
              cF2 for Filter of X2,
             cBa1 for basis of cF1,
             cBa2 for basis of cF2;

theorem Th37:
  <. Frechet_Filter(NAT),Frechet_Filter(NAT).)
    is_filter-finer_than Frechet_Filter([:NAT,NAT:])
  proof
    now
      let x be object;
      assume
A1:   x in Frechet_Filter([:NAT,NAT:]);
      x in the set of all [:NAT,NAT:] \ A where
      A is finite Subset of [:NAT,NAT:] by A1,CARDFIL2:51;
      then consider A be finite Subset of [:NAT,NAT:] such that
A2:   x = [:NAT,NAT:] \ A;
      reconsider y = x as Subset of [:NAT,NAT:] by A2;
      consider m,n such that
A3:   A c= [:Segm m,Segm n:] by Th16;
A4:   [:NAT,NAT:] \ [:Segm m,Segm n:] c= y by A2,A3,XBOOLE_1:34;
      [: NAT \ Segm m , NAT \ Segm n:] c= [:NAT,NAT:] \ [:Segm m,Segm n:]
        by Th10; then
A5:   [: NAT \ Segm m , NAT \ Segm n:] c= y by A4;
      NAT \ Segm m is Element of base_of_frechet_filter &
        NAT \ Segm n is Element of base_of_frechet_filter by Th21;
      then [:NAT \ Segm m ,NAT \ Segm n :] in
        [: base_of_frechet_filter,base_of_frechet_filter :];
      hence x in <.Frechet_Filter(NAT),Frechet_Filter(NAT).)
        by Th35,A5,CARDFIL2:def 8;
    end;
    then Frechet_Filter([:NAT,NAT:]) c=
    <. Frechet_Filter(NAT),Frechet_Filter(NAT).);
    hence thesis;
  end;
