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
  for M being non empty MetrSpace, p being Point of M,
      x being Point of TopSpaceMetr(M),
      s being Function of [:NAT,NAT:], TopSpaceMetr(M),
      s2 being Function of [:NAT,NAT:], M st
  x = p & s = s2 holds
  x in lim_filter(s,<. Frechet_Filter(NAT),Frechet_Filter(NAT).)) iff
  for m being non zero Nat ex n being Nat st for n1,n2 being Nat st
  n <= n1 & n <= n2 holds s2.(n1,n2) in
  {q where q is Point of M: dist(p,q) < 1/m} by Th57;
