
theorem

:: 1.11. THEOREM, (1) <=> (3+), p. 147
  for T being Lawson complete continuous TopLattice
  for S being meet-inheriting full non empty SubRelStr of T
  for X being Subset of T st X = the carrier of S & Top T in X holds
  X is closed iff
  for N being net of T st N is_eventually_in X holds lim_inf N in X
proof
  let T be Lawson complete continuous TopLattice;
  let S be meet-inheriting full non empty SubRelStr of T;
  let X be Subset of T such that
A1: X = the carrier of S and
A2: Top T in X;
  hereby
    assume X is closed;
    then S is infs-inheriting directed-sups-inheriting full non empty
    SubRelStr of T by A1,A2,Th48,Th49;
    hence
    for N being net of T st N is_eventually_in X holds lim_inf N in X
    by A1,Th51;
  end;
  assume for N being net of T st N is_eventually_in X holds lim_inf N in X;
  then 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 by A1,Th27;
  then S is infs-inheriting directed-sups-inheriting by A1,A2,Th52,Th53;
  then ex X being Subset of T st X = the carrier of S & X is closed by Th50;
  hence thesis by A1;
end;
