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

theorem Th39:
  for T being non empty TopSpace,s being sequence of T,
  x being Point of T,
  B being basis of BOOL2F NeighborhoodSystem x holds
  x in lim_filter(s,Frechet_Filter(NAT)) iff
  B is_coarser_than s.:base_of_frechet_filter
  proof
    let T be non empty TopSpace, s be sequence of T,
    x be Point of T, B be basis of BOOL2F NeighborhoodSystem x;
    set F=filter_image(s,Frechet_Filter(NAT));
    hereby
      assume x in lim_filter(s,Frechet_Filter(NAT));
      then consider x0 be Element of T such that
A1:   x=x0 and
A2:   F is_filter-finer_than NeighborhoodSystem x0;
      BOOL2F NeighborhoodSystem x is_filter-coarser_than F by A1,A2;
      hence B is_coarser_than s.:base_of_frechet_filter
      by Th19,Th27;
    end;
    assume
A3: B is_coarser_than s.:base_of_frechet_filter;
    BOOL2F NeighborhoodSystem x is_filter-coarser_than F
    by A3,Th19,Th27;
    then F is_filter-finer_than NeighborhoodSystem x;
    hence x in lim_filter(s,Frechet_Filter(NAT));
  end;
