reserve T for TopSpace;
reserve T for non empty TopSpace;
reserve F for Subset-Family of T;
reserve T for non empty TopSpace;

theorem Th70:
  for F being Subset-Family of T holds (for B being Subset of T st
B in F holds (meet F) /\ (Cl Int(meet F)) c= B) & (F = {} or for A being Subset
  of T st A is condensed holds (for B being Subset of T st B in F holds A c= B)
  implies A c= (meet F) /\ (Cl Int(meet F)))
proof
  let F be Subset-Family of T;
  thus for B being Subset of T st B in F holds (meet F) /\ (Cl Int(meet F)) c=
  B
  proof
    let B be Subset of T;
    assume B in F;
    then
A1: meet F c= B by SETFAM_1:3;
    (meet F) /\ (Cl Int(meet F)) c= meet F by XBOOLE_1:17;
    hence thesis by A1;
  end;
  thus F = {} or for A being Subset of T st A is condensed holds (for B being
  Subset of T st B in F holds A c= B) implies A c= (meet F) /\ (Cl Int(meet F))
  proof
    assume
A2: F <> {};
    let A be Subset of T;
    assume A is condensed;
    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;
    then for P be 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 thesis by A4,XBOOLE_1:19;
  end;
end;
