Open Rectangles Closed under Intersection

Theorem
Let $\left(({\mathbf a .. \mathbf b}\right))$ and $\left(({\mathbf c .. \mathbf d}\right))$ be open $n$-rectangles.

Then $\left(({\mathbf a .. \mathbf b}\right)) \cap \left(({\mathbf c .. \mathbf d}\right))$ is also an open $n$-rectangle.