reserve T for TopSpace;

theorem
  for F being Subset-Family of T holds meet(Int F) c= meet F
proof
  let F be Subset-Family of T;
A1: for A being set st A in F holds meet(Int F) c= A
  proof
    let A be set;
    assume
A2: A in F;
    then reconsider A0 = A as Subset of T;
    set C = Int A0;
    C in {P where P is Subset of T : ex Q being Subset of T st P = Int Q &
    Q in F} by A2;
    then C in Int F by Th16;
    then
A3: meet(Int F) c= C by SETFAM_1:3;
    C c= A0 by TOPS_1:16;
    hence thesis by A3;
  end;
  now
    per cases;
    suppose
      F = {};
      hence thesis by Th18;
    end;
    suppose
      F <> {};
      hence thesis by A1,SETFAM_1:5;
    end;
  end;
  hence thesis;
end;
