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

theorem
  meet(Cl Int F) c= meet(Cl Int Cl F)
proof
  now
    per cases;
    suppose
      F = {};
      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(Cl Int 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;
          consider D being Subset of T such that
A6:       B = Int D and
A7:       D in Cl F by A5,Def1;
          consider E being Subset of T such that
A8:       D = Cl E and
A9:       E in F by A7,PCOMPS_1:def 2;
          Int E in Int F by A9,Def1;
          then Cl Int E in Cl Int F by PCOMPS_1:def 2;
          then
A10:      x in Cl Int E by A2,SETFAM_1:def 1;
          Cl Int E c= Cl Int Cl E by Th2;
          hence thesis by A4,A6,A8,A10;
        end;
        hence x in meet(Cl Int Cl F) by A1,SETFAM_1:def 1;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
