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;
reserve T for non empty TopSpace,
        s for Function of [:NAT,NAT:], the carrier of T,
        M for Subset of the carrier of T;
reserve cF3,cF4 for Filter of the carrier of T;

theorem Th41:
  s"(M) in Frechet_Filter([:NAT,NAT:]) iff
    ex A being finite Subset of [:NAT,NAT:] st s"(M) = [:NAT,NAT:] \ A
  proof
    hereby
      assume s"(M) in Frechet_Filter([:NAT,NAT:]);
      then s"(M) in the set of all [:NAT,NAT:] \ A where
        A is finite Subset of [:NAT,NAT:] by CARDFIL2:51;
      hence ex A be finite Subset of [:NAT,NAT:] st s"(M) = [:NAT,NAT:] \ A;
    end;
    assume ex A be finite Subset of [:NAT,NAT:] st s"(M) = [:NAT,NAT:] \ A;
    then s"(M) in the set of all [:NAT,NAT:] \ A
      where A is finite Subset of [:NAT,NAT:];
    hence s"(M) in Frechet_Filter([:NAT,NAT:]) by CARDFIL2:51;
  end;
