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

theorem
  meet(Cl Int Cl F) c= meet(Cl F)
proof
    per cases;
    suppose
A1:   F = {};
      then Cl F = {} by Th9;
      hence thesis by A1,Th18;
    end;
    suppose
      F <> {};
      then
A2:   Cl F <> {} by Th9;
        let x be object;
        assume
A3:     x in meet(Cl Int Cl F);
        for A being set st A in Cl F holds x in A
        proof
          let A be set;
          assume
A4:       A in Cl F;
          then reconsider A as Subset of T;
          consider B being Subset of T such that
A5:       A = Cl B and
A6:       B in F by A4,PCOMPS_1:def 2;
          Cl B in Cl F by A6,PCOMPS_1:def 2;
          then Int Cl B in Int Cl F by Def1;
          then Cl Int Cl B in Cl Int Cl F by PCOMPS_1:def 2;
          then
A7:       x in Cl Int Cl B by A3,SETFAM_1:def 1;
          Cl Int Cl B c= Cl B by TDLAT_1:3;
          hence thesis by A5,A7;
        end;
        hence x in meet(Cl F) by A2,SETFAM_1:def 1;
    end;
end;
