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

theorem Th41:
  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_filter(s,Frechet_Filter(NAT)) iff
  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
  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;
    x in lim_filter(s,Frechet_Filter(NAT)) iff
    B is_coarser_than s.:base_of_frechet_filter by Th39;
    hence thesis by Th40;
  end;
