
theorem Th53:

:: 1.11. THEOREM, (3) => (2b), p. 147
  for T being Lawson complete continuous TopLattice
  for S being full non empty SubRelStr of T
  st 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 directed-sups-inheriting
proof
  let T be Lawson complete continuous TopLattice;
  let S be full non empty SubRelStr of T;
  set X = the carrier of S;
  assume
A1: for N being net of T st rng the mapping of N c= X holds lim_inf N in X;
  let Y be directed Subset of S;
  assume Y <> {};
  then reconsider F = Y as non empty directed Subset of T by YELLOW_2:7;
  assume ex_sup_of Y,T;
  the mapping of Net-Str F = id F by Th32;
  then
A2: rng the mapping of Net-Str F = Y;
  lim_inf Net-Str F = sup F by WAYBEL17:10;
  hence thesis by A1,A2;
end;
