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;
