Show that the half-open intervals \(\mathscr{J}\) in \(\mathbb{R}^{n}\) are stable under finite intersections. [ check that \(\left.\mathrm{X}_{i=1}^{n}\left[a_{i}, b_{i}ight) \cap \mathrm{X}_{i=1}^{n}\left[a_{i}^{\prime}, b_{i}^{\prime}ight)=\mathrm{X}_{i=1}^{n}\left[a_{i} \vee a_{i}^{\prime}, b_{i} \wedge b_{i}^{\prime}ight).ight]\)