
theorem Th26:
  for R being /\-complete Semilattice, N being net of R,
  V being upper Subset of R holds
  inf_net N is_eventually_in V implies N is_eventually_in V
proof
  let R be /\-complete Semilattice, N be net of R, V be upper Subset of R;
  consider f being Function of N,R such that
A1: inf_net N = N*'f and
A2: for i being Element of N holds f.i = "/\" ({N.k where k is Element
  of N: k >= i},R)
  by Def4;
A3: the RelStr of inf_net N = the RelStr of N by A1,Def3;
  assume inf_net N is_eventually_in V;
  then consider i being Element of inf_net N such that
A4: for j being Element of inf_net N st i <= j holds (inf_net N).j in V;
  consider j0 being Element of inf_net N such that
A5: i <= j0 and i <= j0 by YELLOW_6:def 3;
A6: (inf_net N).j0 in V by A4,A5;
  reconsider j9 = j0 as Element of N by A3;
  take j9;
  let j be Element of N such that
A7: j9 <= j;
  defpred P[Element of N] means $1 >= j9;
  deffunc F(Element of N) = N.$1;
  set E = {F(k) where k is Element of N: P[k]};
  E is Subset of R from DOMAIN_1:sch 8;
  then reconsider E as Subset of R;
  the mapping of (inf_net N) = f by A1,Def3;
  then
A8: (inf_net N).j0 = "/\"(E,R) by A2;
  N.j in E by A7;
  then "/\"(E,R) <= N.j by Th8;
  hence thesis by A6,A8,WAYBEL_0:def 20;
end;
