theorem Th46:
  for x being Point of T holds
  x in lim_filter(s,Frechet_Filter([:NAT,NAT:])) iff
  for A being a_neighborhood of x holds [:NAT,NAT:] \ s"(A) is finite
  proof
    let x be Point of T;
    hereby
      assume
A1:   x in lim_filter(s,Frechet_Filter([:NAT,NAT:]));
      now
        let A be a_neighborhood of x;
        ex B being finite Subset of [:NAT,NAT:] st
          s"(A) = [:NAT,NAT:] \ B by A1,Th45;
        hence [:NAT,NAT:] \ s"(A) is finite by Th1;
      end;
      hence for A be a_neighborhood of x holds [:NAT,NAT:] \ s"(A) is finite;
    end;
    assume
A2: for A be a_neighborhood of x holds [:NAT,NAT:] \ s"(A) is finite;
    now
      let A be a_neighborhood of x;
A3:   dom s = [:NAT,NAT:] by FUNCT_2:def 1;
      [:NAT,NAT:] \ s"(A) is finite by A2;
      hence ex B being finite Subset of [:NAT,NAT:] st
        s"(A) = [:NAT,NAT:] \ B by A3,RELAT_1:132,Th2;
    end;
    hence x in lim_filter(s,Frechet_Filter([:NAT,NAT:])) by Th45;
  end;
