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

theorem
  for X be non empty set, s be sequence of X,
  B be filter_base of 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 by Th20;
