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

assume F is closed ; :: thesis:
then meet F is closed by TOPS_2:22;
then A1: Cl (meet F) = meet F by PRE_TOPC:22;
let B be Subset of T; :: thesis:
A2: Cl (Int (meet F)) c= Cl (meet F) by PRE_TOPC:19, TOPS_1:16;
assume B in F ; :: thesis:
then meet F c= B by SETFAM_1:3;
hence Cl (Int (meet F)) c= B by A2, A1; :: thesis:

end;

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

assume A3: F <> {} ; :: thesis:
let A be Subset of T; :: thesis:
assume A4: A is closed_condensed ; :: thesis:
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 A c= meet F by A3, SETFAM_1:5;
then A5: Int A c= Int (meet F) by TOPS_1:19;
A = Cl (Int A) by A4, TOPS_1:def 7;
hence A c= Cl (Int (meet F)) by A5, PRE_TOPC:19; :: thesis:

end;