reserve T for TopSpace;

theorem Th13:
  for F being Subset-Family of T holds meet F c= meet(Cl F)
proof
  let F be Subset-Family of T;
A1: for A being set st A in Cl F holds meet F c= A
  proof
    let A be set;
    assume
A2: A in Cl F;
    then reconsider A0 = A as Subset of T;
    consider B being Subset of T such that
A3: A0 = Cl B and
A4: B in F by A2,PCOMPS_1:def 2;
A5: B c= A0 by A3,PRE_TOPC:18;
    meet F c= B by A4,SETFAM_1:3;
    hence thesis by A5;
  end;
  now
    per cases;
    suppose
      Cl F = {};
      hence thesis by Th9;
    end;
    suppose
      Cl F <> {};
      hence thesis by A1,SETFAM_1:5;
    end;
  end;
  hence thesis;
end;
