let L be complete LATTICE; :: thesis: for A being Subset of L st A is lower holds
( A ` in xi L iff A is closed_under_directed_sups )
let A be Subset of L; :: thesis: ( A is lower implies ( A ` in xi L iff A is closed_under_directed_sups ) )
set T = the Scott TopAugmentation of L;
assume A1: A is lower ; :: thesis: ( A ` in xi L iff A is closed_under_directed_sups )
then reconsider A1 = A as lower Subset of L ;
A2: RelStr(# the carrier of L, the InternalRel of L #) = RelStr(# the carrier of the Scott TopAugmentation of L, the InternalRel of the Scott TopAugmentation of L #) by YELLOW_9:def 4;
then reconsider A9 = A as Subset of the Scott TopAugmentation of L ;
reconsider A9 = A9 as Subset of the Scott TopAugmentation of L ;
A3: A1 ` is upper ;
thus ( A ` in xi L implies A is closed_under_directed_sups ) :: thesis: ( A is closed_under_directed_sups implies A ` in xi L )
proof
assume A ` in xi L ; :: thesis: A is closed_under_directed_sups
then A9 ` in sigma L by A3, A2, Th29;
then A9 ` in the topology of the Scott TopAugmentation of L 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 A is closed_under_directed_sups by A2, YELLOW12:20; :: thesis: verum
end;
assume A is closed_under_directed_sups ; :: thesis: A ` in xi L
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 the Scott TopAugmentation of L by PRE_TOPC:def 2;
then A ` in sigma L by A2, YELLOW_9:51;
hence A ` in xi L by Th28; :: thesis: verum