
theorem ::3.3 Proposition (ii)
  for L being complete LATTICE , A being Subset of L st A is lower holds
  A` in xi L iff A is closed_under_directed_sups
proof
  let L be complete LATTICE, A be Subset of L;
  set T = the Scott TopAugmentation of L;
  assume
A1: A is lower;
  then reconsider A1=A as lower Subset of L;
A2: the RelStr of L = the RelStr of T by YELLOW_9:def 4;
  then reconsider A9=A as Subset of T;
  reconsider A9 as Subset of T;
A3: A1` is upper;
  thus A` in xi L implies A is closed_under_directed_sups
  proof
    assume A` in xi L;
    then A9` in sigma L by A3,A2,Th29;
    then A9` in the topology of T by YELLOW_9:51;
    then A9` is open by PRE_TOPC:def 2;
    then A9 is closed by TOPS_1:3;
    then A9 is directly_closed by WAYBEL11:7;
    hence thesis by A2,YELLOW12:20;
  end;
  assume A is closed_under_directed_sups;
  then
A4: A9 is directly_closed by A2,YELLOW12:20;
  A9 is lower by A1,A2,WAYBEL_0:25;
  then A9 is closed by A4,WAYBEL11:7;
  then A9` is open by TOPS_1:3;
  then A9` in the topology of T by PRE_TOPC:def 2;
  then A` in sigma L by A2,YELLOW_9:51;
  hence thesis by Th28;
end;
