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

theorem
  meet(Int F) c= meet(Int Cl Int F)
proof
    per cases;
    suppose
A1:   F = {};
      then Int F = {} by Th18;
      hence thesis by A1,Th9;
    end;
    suppose
      F <> {};
      then Int F <> {} by Th18;
      then Cl Int F <> {} by Th9;
      then
A2:   Int Cl Int F <> {} by Th18;
        let x be object;
        assume
A3:     x in meet(Int F);
        for A being set st A in Int Cl Int F holds x in A
        proof
          let A be set;
          assume
A4:       A in Int Cl Int F;
          then reconsider A as Subset of T;
          consider E being Subset of T such that
A5:       A = Int E and
A6:       E in Cl Int F by A4,Def1;
          consider B being Subset of T such that
A7:       E = Cl B and
A8:       B in Int F by A6,PCOMPS_1:def 2;
          consider D being Subset of T such that
A9:       B = Int D and
A10:      D in F by A8,Def1;
          Int D in Int F by A10,Def1;
          then
A11:      x in Int D by A3,SETFAM_1:def 1;
          Int D c= Int Cl Int D by TDLAT_1:4;
          hence thesis by A5,A7,A9,A11;
        end;
        hence x in meet(Int Cl Int F) by A2,SETFAM_1:def 1;
    end;
end;
