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

theorem
  for T being non empty TopSpace,s being sequence of the carrier of T,
  x being Point of T,
  B being basis of BOOL2F NeighborhoodSystem x holds
  x in lim_f s iff for b be Element of B holds
  sequence_to_net(s) is_eventually_in b
  proof
    let T be non empty TopSpace,s be sequence of the carrier of T,
    x be Point of T,
    B be basis of BOOL2F NeighborhoodSystem x;
    hereby
      assume x in lim_f s;
      then for b be Element of B ex i be Element of OrderedNAT st
      for j be Element of OrderedNAT st i <=j holds s.j in b by Th42;
      hence for b be Element of B holds sequence_to_net(s) is_eventually_in b
      by Th44;
    end;
    assume for b be Element of B holds sequence_to_net(s) is_eventually_in b;
    then (for b be Element of B
    ex i be Element of sequence_to_net(s) st
    for j be Element of sequence_to_net(s) st
    i <=j holds (sequence_to_net(s)).j in b) by Th43;
    then for b be Element of B ex i be Element of OrderedNAT st
    for j be Element of OrderedNAT st i <=j holds s.j in b by Th44;
    hence x in lim_f s by Th42;
  end;
