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

theorem
  for T being non empty TopSpace, F being Filter of the carrier of T holds
  Lim a_net F2BOOL(F,T)=lim_filter F
  proof
    let T be non empty TopSpace,F be Filter of the carrier of T;
    now
      hereby let x be object; assume
A1:     x in Lim a_net F2BOOL(F,T);
        then reconsider x0=x as Point of T;
        F is_filter-finer_than NeighborhoodSystem x0
        by A1,YELLOW19:3,YELLOW19:17;
        hence x in lim_filter F;
      end;
      let x be object; assume
A2:   x in lim_filter F;
      then reconsider x0=x as Point of T;
      consider x1 be Point of T such that
A3:   x0=x1 and
A4:   F is_filter-finer_than NeighborhoodSystem x1 by A2;
      thus x in Lim a_net F2BOOL(F,T) by A3,A4,YELLOW19:3,YELLOW19:17;
    end;
    then Lim a_net F2BOOL(F,T) c= lim_filter F &
    lim_filter F c= Lim a_net F2BOOL(F,T);
    hence thesis;
  end;
