
theorem
  for M being non empty MetrSpace,
      L being non empty transitive reflexive RelStr,
      f being Function of [#]L,the carrier of TopSpaceMetr(M),
      x being Point of TopSpaceMetr(M) st
    [#]L is directed holds
      x in lim_f f
     iff
      for b being Element of Balls(x) ex n being Element of L st
      for m being Element of L st n <= m holds f.m in b
  proof
    let M be non empty MetrSpace,
    L be non empty transitive reflexive RelStr,
    f be Function of [#]L,the carrier of TopSpaceMetr(M),
    x be Point of TopSpaceMetr(M);
    assume
A1: [#]L is directed;
    Balls(x) is basis of BOOL2F NeighborhoodSystem x by Th5;
    hence thesis by A1,CARDFIL2:84;
  end;
