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 Th49:
  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 A being finite Subset of [:NAT,NAT:] st
    s"(B) = [:NAT,NAT:] \ A
  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:]);
      hereby
        let B be Element of cB;
        B is a_neighborhood of x by YELLOW19:2;
        hence ex A being finite Subset of [:NAT,NAT:] st
          s"(B) = [:NAT,NAT:] \ A by A1,Th45;
      end;
    end;
    assume
A2: for B being Element of cB holds ex A being finite Subset of [:NAT,NAT:]
      st s"(B) = [:NAT,NAT:] \ A;
    now
      let A be a_neighborhood of x;
A3:   A is Element of BOOL2F NeighborhoodSystem x by YELLOW19:2;
      cB is filter_basis;
      then consider B be Element of cB such that
A4:   B c= A by A3;
      ex C be finite Subset of [:NAT,NAT:] st s"(B) = [:NAT,NAT:] \ C by A2;
      hence [:NAT,NAT:] \ s"(A) is finite by A4,RELAT_1:145,Th1;
    end;
    hence x in lim_filter(s,Frechet_Filter([:NAT,NAT:])) by Th46;
  end;
