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 Th43:
  filter_image(s,Frechet_Filter([:NAT,NAT:])) = {M where M is Subset of
    the carrier of T: ex A being finite Subset of [:NAT,NAT:] st
    s"(M) = [:NAT,NAT:] \ A}
  proof
    set X = {M where M is Subset of the carrier of T: s"(M) in
      Frechet_Filter([:NAT,NAT:])},
    Y = {M where M is Subset of the carrier of T:
      ex A be finite Subset of [:NAT,NAT:] st s"(M) = [:NAT,NAT:] \ A};
    X = Y
    proof
      now
        let x be object;
        assume x in X;
        then consider M be Subset of the carrier of T such that
A1:     x = M and
A2:     s"(M) in Frechet_Filter([:NAT,NAT:]);
        ex A be finite Subset of [:NAT,NAT:] st s"(M) = [:NAT,NAT:] \ A
          by Th41,A2;
        hence x in Y by A1;
      end;
      then
A3:   X c= Y;
      now
        let x be object;
        assume x in Y;
        then consider M be Subset of the carrier of T such that
A4:     x = M and
A5:     ex A be finite Subset of [:NAT,NAT:] st s"(M) = [:NAT,NAT:] \ A;
        s"(M) in Frechet_Filter([:NAT,NAT:]) by A5,Th41;
        hence x in X by A4;
      end;
      then Y c= X;
      hence thesis by A3;
    end;
    hence thesis;
  end;
