reserve T for TopSpace;
reserve T for non empty TopSpace;
reserve F for Subset-Family of T;

theorem
  meet(Int Cl F) c= meet(Cl Int Cl F)
proof
  now
    per cases;
    suppose
      F = {};
      then Cl F = {} by Th9;
      then Int Cl F = {} by Th18;
      hence thesis by Th9;
    end;
    suppose
      F <> {};
      then Cl F <> {} by Th9;
      then Int Cl F <> {} by Th18;
      then
A1:   Cl Int Cl F <> {} by Th9;
      now
        let x be object;
        assume
A2:     x in meet(Int Cl F);
        for A being set st A in Cl Int Cl F holds x in A
        proof
          let A be set;
          assume
A3:       A in Cl Int Cl F;
          then reconsider A as Subset of T;
          consider B being Subset of T such that
A4:       A = Cl B and
A5:       B in Int Cl F by A3,PCOMPS_1:def 2;
A6:       B c= Cl B by PRE_TOPC:18;
          x in B by A2,A5,SETFAM_1:def 1;
          hence thesis by A4,A6;
        end;
        hence x in meet(Cl Int Cl F) by A1,SETFAM_1:def 1;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
