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;
