reserve
  X for non empty set,
  FX for Filter of X,
  SFX for Subset-Family of X;

theorem
  for T being non empty TopSpace,x being Point of T,
  F be Filter of the carrier of T holds x is_a_convergence_point_of F,T iff
  x in lim_filter F
  proof
    let T be non empty TopSpace,x be Point of T,
    F be Filter of the carrier of T;
    consider F2 be proper Filter of BoolePoset the carrier of T such that
A1: F=F2 by Th37;
    F is_filter-finer_than NeighborhoodSystem x iff
    x in {x where x is Point of T: F is_filter-finer_than NeighborhoodSystem x}
    proof
      thus F is_filter-finer_than NeighborhoodSystem x implies
      x in {x where x is Point of T: F is_filter-finer_than
      NeighborhoodSystem x};
      assume x in {x where x is Point of T: F is_filter-finer_than
      NeighborhoodSystem x};
      then consider x0 be Point of T such that
A2:   x=x0 and
A3:   F is_filter-finer_than NeighborhoodSystem x0;
      thus F is_filter-finer_than NeighborhoodSystem x by A2,A3;
    end;
    hence thesis by A1,YELLOW19:3;
  end;
