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 Th54:
  for x being Point of T, cB being basis of BOOL2F NeighborhoodSystem x holds
  x in lim_filter(s,<. Frechet_Filter(NAT),Frechet_Filter(NAT).)) iff
  for B being Element of cB holds ex n being Nat st for n1,n2 being Nat st
  n <= n1 & n <= n2 holds s.(n1,n2) in B
  proof
    let x be Point of T, cB be basis of BOOL2F NeighborhoodSystem x;
    (for B being Element of cB holds ex n being Nat st
      s.:(square-uparrow n) c= B) iff
      (for B being Element of cB holds ex n being Nat st
       for n1,n2 being Nat st n <= n1 & n <= n2 holds s.(n1,n2) in B)
    proof
      hereby
        assume
A1:     for B being Element of cB holds ex n being Nat st
          s.:(square-uparrow n) c= B;
        hereby
          let B be Element of cB;
          consider n0 being Nat such that
A2:       s.:(square-uparrow n0) c= B by A1;
          take n0;
          thus for n1,n2 be Nat st n0 <= n1 & n0 <= n2 holds s.(n1,n2) in B
            by A2,Th53;
        end;
      end;
      assume
A3:   for B being Element of cB holds ex n being Nat st
        for n1,n2 being Nat st n <= n1 & n <= n2 holds s.(n1,n2) in B;
      hereby
        let B be Element of cB;
        consider n0 be Nat such that
A4:     for n1,n2 being Nat st n0 <= n1 & n0 <= n2 holds s.(n1,n2) in B by A3;
        thus ex n being Nat st s.:(square-uparrow n) c= B
        proof
          take n0;
          let x be object;
          assume x in s.:(square-uparrow n0);
          then ex n1,n2 be Nat st n0 <= n1 & n0 <= n2 & x = s.(n1,n2) by Th53;
          hence thesis by A4;
        end;
      end;
    end;
    hence thesis by Th50;
  end;
