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

theorem
  meet(Int Cl Int F) c= meet(Cl Int F)
proof
    per cases;
    suppose
      F = {};
      then Int F = {} by Th18;
      then Cl Int F = {} by Th9;
      hence thesis by Th18;
    end;
    suppose
      F <> {};
      then Int F <> {} by Th18;
      then
A1:   Cl Int F <> {} by Th9;
      now
        let x be object;
        assume
A2:     x in meet(Int Cl Int F);
        for A being set st A in Cl Int F holds x in A
        proof
          let A be set;
          assume
A3:       A in Cl Int F;
          then reconsider A as Subset of T;
          consider E being Subset of T such that
A4:       A = Cl E and
A5:       E in Int F by A3,PCOMPS_1:def 2;
          consider B being Subset of T such that
A6:       E = Int B and
A7:       B in F by A5,Def1;
          Int B in Int F by A7,Def1;
          then Cl Int B in Cl Int F by PCOMPS_1:def 2;
          then Int Cl Int B in Int Cl Int F by Def1;
          then
A8:       x in Int Cl Int B by A2,SETFAM_1:def 1;
          Int Cl Int B c= Cl Int B by Th1;
          hence thesis by A4,A6,A8;
        end;
        hence x in meet(Cl Int F) by A1,SETFAM_1:def 1;
      end;
      hence thesis;
    end;
end;
