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 Th33:
  ex cB being basis of Frechet_Filter(NAT) st
  cB = base_of_frechet_filter & [: cB, cB :] is
  basis of <. Frechet_Filter(NAT),Frechet_Filter(NAT) .)
  proof
    reconsider bff = base_of_frechet_filter as basis of Frechet_Filter(NAT)
      by CARDFIL2:56;
    [: bff,bff :] is basis of <. Frechet_Filter(NAT),Frechet_Filter(NAT).);
    hence thesis;
  end;
