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

theorem Th40:
  for T being non empty TopSpace,s being sequence of [#]T, x being Point of T,
  B being basis of BOOL2F NeighborhoodSystem x holds
  B is_coarser_than s.:base_of_frechet_filter 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 [#]T,
    x be Point of T,
    B be basis of BOOL2F NeighborhoodSystem x;
    reconsider B as filter_base of [#]T by Th09;
    B is_coarser_than s.:base_of_frechet_filter 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 by Th20;
    hence thesis;
  end;
