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

theorem
  for T being non empty TopSpace,
      L being non empty transitive reflexive RelStr,
  f being Function of [#]L,the carrier of T, x being Point of T,
  B being basis of BOOL2F NeighborhoodSystem x st
  [#]L is directed holds
  x in lim_f f iff
  for b be Element of B ex i be Element of L st
  for j be Element of L st i <=j holds f.j in b
  proof
    let T be non empty TopSpace, L be non empty transitive reflexive RelStr,
    f be Function of [#]L,the carrier of T, x be Point of T,
    B be basis of BOOL2F NeighborhoodSystem x such that
A1: [#]L is directed;
    hereby
      assume x in lim_f f;
      then consider x0 be Element of T such that
A2:   x=x0 and
A3:   filter_image(f,Tails_Filter L) is_filter-finer_than
      NeighborhoodSystem x0;
      BOOL2F NeighborhoodSystem x is_filter-coarser_than
      filter_image(f,Tails_Filter L) by A2,A3;
      then
A4:   B is_coarser_than f.:#(Tails L) by A1,Th19;
      reconsider B1=B as filter_base of [#]T by Th09;
      for b be Element of B1 ex i be Element of L st
      for j be Element of L st i <=j holds f.j in b by A1,A4,Th20;
      hence for b be Element of B ex i be Element of L st
      for j be Element of L st i <=j holds f.j in b;
    end;
    assume
A5: for b be Element of B ex i be Element of L st
    for j be Element of L st i <=j holds f.j in b;
    reconsider B1=B as filter_base of [#]T by Th09;
    for b be Element of B1 ex i be Element of L st
    for j be Element of L st i <=j holds f.j in b by A5;
    then B is_coarser_than f.:#(Tails L) by A1,Th20;
    then BOOL2F NeighborhoodSystem x is_filter-coarser_than
    filter_image(f,Tails_Filter L) by A1,Th19;
    then x is Element of T & filter_image(f,Tails_Filter L)
    is_filter-finer_than NeighborhoodSystem x;
    hence x in lim_f f;
  end;
