
theorem
  for X being non empty LinearTopSpace,
      s being sequence of the carrier of X,
      x being Point of X,
      V being local_base of X,
      B being Subset-Family of X st
  B={x+U where U is Subset of X:U in V & U is a_neighborhood of 0.X} holds
  x in lim_f s
    iff
  for v being Element of B holds ex i being Nat st
  for j being Nat st i <=j holds s.j in v
  proof
    let X be non empty LinearTopSpace,
    s be sequence of the carrier of X,
    x be Point of X, V be local_base of X,B be
    Subset-Family of X;
    assume B={x+U where U is Subset of X:U in V &
    U is a_neighborhood of 0.X};
    then B is basis of BOOL2F NeighborhoodSystem x by Th10;
    hence x in lim_f s iff
    for b be Element of B ex i be Nat st
    for j be Nat st i <=j holds s.j in b by CARDFIL2:97;
  end;
