reserve x for set;

theorem Th16:
  for L being complete LATTICE, F being ultra Filter of BoolePoset
  [#]L holds for M being subnet of a_net F holds lim_inf F = lim_inf M
proof
  let L be complete LATTICE, F be ultra Filter of BoolePoset [#]L, M be subnet
  of a_net F;
  lim_inf F = lim_inf a_net F & for p being greater_or_equal_to_id
  Function of a_net F, a_net F holds lim_inf F >= inf ((a_net F) * p) by Th13
,Th15;
  hence thesis by WAYBEL28:14;
end;
