reserve T for TopSpace;

theorem
  for F being Subset-Family of T, A being Subset of T st A in F holds
  meet(Cl F) c= Cl A & Cl A c= union(Cl F)
proof
  let F be Subset-Family of T, A be Subset of T;
  assume A in F;
  then
  Cl A in {P where P is Subset of T : ex B being Subset of T st P = Cl B &
  B in F};
  then
A1: Cl A in Cl F by Th7;
  hence meet(Cl F) c= Cl A by SETFAM_1:3;
  thus thesis by A1,ZFMISC_1:74;
end;
