
theorem Th12:
  for I being set,
      G being TopaddGroup,
      f being Function of [#]OrderedFIN I,the carrier of G,
      x being Point of G,
      B being basis of BOOL2F NeighborhoodSystem x holds
  x in lim_f f
    iff
  for b being Element of B ex i being Element of OrderedFIN I st
  for j being Element of OrderedFIN I st i <= j holds f.j in b
  proof
    let I be set,G be TopaddGroup,
    f be Function of [#]OrderedFIN I,the carrier of G,
    x be Point of G,
    B be basis of BOOL2F NeighborhoodSystem x;
    [#]OrderedFIN I is directed by Th4;
    hence thesis by CARDFIL2:84;
  end;
