
theorem Th51:

:: 1.11. THEOREM, (2) => (3+), p. 147
  for T being Lawson complete continuous TopLattice
  for S being infs-inheriting directed-sups-inheriting full non empty
  SubRelStr of T for N being net of T st N is_eventually_in the carrier of S
  holds lim_inf N in the carrier of S
proof
  let T be Lawson complete continuous TopLattice;
  let S be infs-inheriting directed-sups-inheriting full non empty
  SubRelStr of T;
  set X = the carrier of S;
  let N be net of T;
  assume N is_eventually_in X;
  then consider a being Element of N such that
  N.a in X and
A1: rng the mapping of N|a c= X by Th42;
  deffunc up(Element of N|a) = {N|a.i where i is Element of N|a: i >= $1};
  reconsider iN = the set of all "/\"(up(j), T)
  where j is Element of N|a
  as directed non empty Subset of T by Th25;
  iN c= X
  proof
    let z be object;
    assume z in iN;
    then consider j being Element of N|a such that
A2: z = "/\"(up(j), T);
    up(j) c= X
    proof
      let u be object;
      assume u in up(j);
      then ex i being Element of N|a st ( u = (N|a).i)&( i >= j);
      then u in rng the mapping of N|a by FUNCT_2:4;
      hence thesis by A1;
    end;
    then reconsider Xj = up(j) as Subset of S;
    ex_inf_of Xj, T by YELLOW_0:17;
    hence thesis by A2,YELLOW_0:def 18;
  end;
  then reconsider jN = iN as non empty Subset of S;
A3: jN is directed by WAYBEL10:23;
  ex_sup_of jN,T by YELLOW_0:17;
  then "\/"(jN,T) in the carrier of S by A3,WAYBEL_0:def 4;
  then lim_inf (N|a) in X by WAYBEL11:def 6;
  hence thesis by Th41;
end;
