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