reserve T for TopSpace;

theorem
  for F,G being Subset-Family of T holds (Cl F) \ (Cl G) c= Cl(F \ G)
proof
  let F,G be Subset-Family of T;
  for X being object holds X in (Cl F) \ (Cl G) implies X in Cl(F \ G)
  proof
    let X be object;
    assume
A1: X in (Cl F) \ (Cl G);
    then reconsider X0 = X as Subset of T;
    X in Cl F by A1,XBOOLE_0:def 5;
    then consider W being Subset of T such that
A2: X0 = Cl W and
A3: W in F by PCOMPS_1:def 2;
    not X in Cl G by A1,XBOOLE_0:def 5;
    then not W in G by A2,PCOMPS_1:def 2;
    then W in F \ G by A3,XBOOLE_0:def 5;
    hence thesis by A2,PCOMPS_1:def 2;
  end;
  hence thesis;
end;
