reserve T for TopSpace;

theorem
  for F,G being Subset-Family of T holds Cl(F /\ G) c= (Cl F) /\ (Cl G)
proof
  let F,G be Subset-Family of T;
  for X being object holds X in Cl(F /\ G) implies X in (Cl F) /\ (Cl G)
  proof
    let X be object;
    assume
A1: X in Cl(F /\ G);
    then reconsider X0 = X as Subset of T;
    consider W being Subset of T such that
A2: X0 = Cl W and
A3: W in F /\ G by A1,PCOMPS_1:def 2;
    W in G by A3,XBOOLE_0:def 4;
    then
A4: X0 in Cl G by A2,PCOMPS_1:def 2;
    W in F by A3,XBOOLE_0:def 4;
    then X0 in Cl F by A2,PCOMPS_1:def 2;
    hence thesis by A4,XBOOLE_0:def 4;
  end;
  hence thesis;
end;
