
theorem
  for R being /\-complete Semilattice, N being net of R,
  V being lower Subset of R holds
  N is_eventually_in V implies inf_net N is_eventually_in V
proof
  let R be /\-complete Semilattice, N be net of R, V be lower 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 N is_eventually_in V;
  then consider i being Element of N such that
A4: for j being Element of N st i <= j holds N.j in V;
  reconsider i9 = i as Element of inf_net N by A3;
  take i9;
  let j be Element of inf_net N such that
A5: i9 <= j;
  reconsider j0 = j as Element of N by A3;
  defpred P[Element of N] means $1 >= j0;
  deffunc F(Element of N) = N.$1;
  set E = {F(k) where k is Element of N: P[k]};
  consider j1 being Element of N such that
A6: j1 >= j0 and j1 >= j0 by YELLOW_6:def 3;
  E is Subset of R from DOMAIN_1:sch 8;
  then reconsider E as Subset of R;
  i <= j0 by A3,A5,YELLOW_0:1;
  then i <= j1 by A6,YELLOW_0:def 2;
  then
A7: N.j1 in V by A4;
  N.j1 in E by A6;
  then
A8: "/\"(E,R) <= N.j1 by Th8;
  the mapping of (inf_net N) = f by A1,Def3;
  then (inf_net N).j = "/\"(E,R) by A2;
  hence thesis by A7,A8,WAYBEL_0:def 19;
end;
