reserve x,y,X for set;

theorem Th3:
  for T being non empty TopSpace, x being Point of T for F being
  upper Subset of BoolePoset [#]T holds x is_a_convergence_point_of F, T iff
  NeighborhoodSystem x c= F
proof
  let T be non empty TopSpace, x be Point of T;
  let F be upper Subset of BoolePoset [#]T;
  hereby
    assume
A1: x is_a_convergence_point_of F, T;
    thus NeighborhoodSystem x c= F
    proof
      let y be object;
      assume y in NeighborhoodSystem x;
      then reconsider y as a_neighborhood of x by Th2;
      x in Int y by CONNSP_2:def 1;
      then
A2:   Int y in F by A1;
      Int y c= y by TOPS_1:16;
      hence thesis by A2,WAYBEL_7:7;
    end;
  end;
  assume
A3: NeighborhoodSystem x c= F;
  let A be Subset of T;
  assume that
A4: A is open and
A5: x in A;
  A is a_neighborhood of x by A4,A5,CONNSP_2:3;
  then A in NeighborhoodSystem x;
  hence thesis by A3;
end;
