
theorem

:: 1.6. PROPOSITION (ii), p. 144
  for T being Lawson complete TopLattice for A being lower Subset of T
  holds A is closed iff A is closed_under_directed_sups
proof
  let T be Lawson complete TopLattice;
  let A be lower Subset of T;
  set S = the Scott TopAugmentation of T;
  hereby
    assume A is closed;
    then A` is open;
    then reconsider mA = A` as property(S) Subset of T by Th36;
    mA` = A;
    hence A is directly_closed;
  end;
  assume A is directly_closed;
  then reconsider dA = A as directly_closed lower Subset of T;
A1: the RelStr of S = the RelStr of T by YELLOW_9:def 4;
  then reconsider mA = dA` as Subset of S;
  mA is upper inaccessible by A1,WAYBEL_0:25,YELLOW_9:47;
  then
A2: mA is open by WAYBEL11:def 4;
  T is TopAugmentation of S by A1,YELLOW_9:def 4;
  then dA` is open by A2,Th37;
  hence thesis;
end;
