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 Th52:
  for x being Point of T, cB being basis of BOOL2F NeighborhoodSystem x holds
  x in lim_filter(s,Frechet_Filter([:NAT,NAT:])) iff for B being Element of cB
  holds ex n,m st s.:([:NAT,NAT:] \ [:Segm n,Segm m:]) c= B
  proof
    let x be Point of T,cB be basis of BOOL2F NeighborhoodSystem x;
    hereby
      assume
A1:   x in lim_filter(s,Frechet_Filter([:NAT,NAT:]));
      now
        let B be Element of cB;
        consider A be finite Subset of [:NAT,NAT:] such that
A2:     s.:([:NAT,NAT:] \ A) c= B by A1,Th51;
        consider n,m such that
A3:     A c= [:Segm n,Segm m:] by Th16;
        [:NAT,NAT:] \ [:Segm n,Segm m:] c= [:NAT,NAT:] \ A by A3,XBOOLE_1:34;
        then s.:([:NAT,NAT:] \ [:Segm n,Segm m:]) c= s.:([:NAT,NAT:] \ A)
          by RELAT_1:123;
        then s.:([:NAT,NAT:] \ [:Segm n,Segm m:]) c= B by A2;
        hence ex n,m st s.:([:NAT,NAT:] \ [:Segm n,Segm m:]) c= B;
      end;
      hence for B being Element of cB holds ex n,m st
        s.:([:NAT,NAT:] \ [:Segm n,Segm m:]) c= B;
    end;
    assume
A4: for B being Element of cB holds ex n,m st
      s.:([:NAT,NAT:] \ [:Segm n,Segm m:]) c= B;
    now
      let B be Element of cB;
      consider n,m such that
A5:   s.:([:NAT,NAT:] \ [:Segm n,Segm m:]) c= B by A4;
      reconsider A = [:Segm n,Segm m:] as finite Subset of [:NAT,NAT:];
      thus ex A be finite Subset of [:NAT,NAT:] st s.:([:NAT,NAT:] \ A) c= B
        by A5;
    end;
    hence x in lim_filter(s,Frechet_Filter([:NAT,NAT:])) by Th51;
  end;
