
theorem Th28:
  for R being order_consistent up-complete /\-complete non empty TopLattice
  for N being net of R, x being Element of R holds
  x <= lim_inf N implies x is_a_cluster_point_of N
proof
  let R be order_consistent up-complete /\-complete non empty TopLattice,
  N be net of R, x be Element of R;
  assume
A1: x <= lim_inf N;
  defpred P[Element of N] means not contradiction;
  deffunc F(Element of N) = "/\"({N.i where i is Element of N:i >= $1}, R);
  set X = {F(j) where j is Element of N: P[j]};
  X is Subset of R from DOMAIN_1:sch 8;
  then reconsider D = X as Subset of R;
  reconsider D as non empty directed Subset of R by Th7;
  sup D = lim_inf N;
  then
A2: sup D = sup inf_net N by Th24;
  let V be a_neighborhood of x;
  for a being Element of R holds downarrow a = Cl {a} by Def2;
  then
A3: Int V is upper by Th21;
  x in Int V by CONNSP_2:def 1;
  then sup D in Int V by A1,A3;
  then reconsider W = Int V as a_neighborhood of (sup D) by CONNSP_2:3;
A4: Int V c= V by TOPS_1:16;
  inf_net N is_eventually_in W by A2,Def2;
  then N is_eventually_in W by A3,Th26;
  then N is_eventually_in V by A4,WAYBEL_0:8;
  hence thesis by YELLOW_6:19;
end;
