
theorem Th48:

:: 1.11. THEOREM, (1) => (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 &
  ex X being Subset of T st X = the carrier of S & X is closed
  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;
  given X being Subset of T such that
A2: X = the carrier of S and
A3: X is closed;
  S is filtered-infs-inheriting
  proof
    let Y be filtered Subset of S;
    assume Y <> {};
    then reconsider F = Y as filtered non empty Subset of T by YELLOW_2:7;
    set N = F opp+id;
    assume ex_inf_of Y, T;
    the mapping of N = id Y by WAYBEL19:27;
    then
A4: rng the mapping of N = Y;
    Lim N = {inf F} by WAYBEL19:43;
    then {inf F} c= Cl X by A2,A4,Th27,WAYBEL19:26;
    then {inf F} c= X by A3,PRE_TOPC:22;
    hence thesis by A2,ZFMISC_1:31;
  end;
  hence thesis by A1,Th16;
end;
