
theorem Th52:

:: 1.11. THEOREM, (3) => (2a), p. 147
  for T being Lawson complete continuous TopLattice
  for S being meet-inheriting full non empty SubRelStr of T
  st Top T in the carrier of S &
  for N being net of T st rng the mapping of N c= the carrier of S
  holds lim_inf N in the carrier of S holds S is infs-inheriting
proof
  let T be Lawson complete continuous TopLattice;
  let S be meet-inheriting full non empty SubRelStr of T such that
A1: Top T in the carrier of S;
  set X = the carrier of S;
  assume
A2: for N being net of T st rng the mapping of N c= X holds lim_inf N in X;
  S is filtered-infs-inheriting
  proof
    let Y be filtered Subset of S;
    assume Y <> {};
    then reconsider F = Y as non empty filtered Subset of T by YELLOW_2:7;
    assume ex_inf_of Y,T;
    the mapping of F opp+id = id F by WAYBEL19:27;
    then
A3: rng the mapping of F opp+id = Y;
    lim_inf (F opp+id) = inf F by Th28;
    hence thesis by A2,A3;
  end;
  hence thesis by A1,Th16;
end;
