let T be non empty TopSpace; :: thesis:
let F be Subset-Family of T; :: thesis:
thus for B being Subset of T st B in F holds
(meet F) /\ (Cl (Int (meet F))) c= B :: thesis:

let B be Subset of T; :: thesis:
assume B in F ; :: thesis:
then A1: meet F c= B by SETFAM_1:3;
(meet F) /\ (Cl (Int (meet F))) c= meet F by XBOOLE_1:17;
hence (meet F) /\ (Cl (Int (meet F))) c= B by A1; :: thesis:

end;

thus ( F = {} or for A being Subset of T st A is condensed & ( for B being Subset of T st B in F holds
A c= B ) holds
A c= (meet F) /\ (Cl (Int (meet F))) ) :: thesis:

assume A2: F <> {} ; :: thesis:
let A be Subset of T; :: thesis:
assume A is condensed ; :: thesis:
then A3: A c= Cl (Int A) by TOPS_1:def 6;
assume for B being Subset of T st B in F holds
A c= B ; :: thesis:
then for P being set st P in F holds
A c= P ;
then A4: A c= meet F by A2, SETFAM_1:5;
then Int A c= Int (meet F) by TOPS_1:19;
then Cl (Int A) c= Cl (Int (meet F)) by PRE_TOPC:19;
then A c= Cl (Int (meet F)) by A3;
hence A c= (meet F) /\ (Cl (Int (meet F))) by A4, XBOOLE_1:19; :: thesis:

end;