Intersection of Quotient Groups

Theorem
Let $N \lhd G$, and let $N \le A \le G, N \le B \le G$.

Let $\alpha$ be the bijection defined as:
 * $\alpha \left({H}\right) = \left\{{h N: h \in H}\right\} \subseteq G / N$

from the proof of the Correspondence Theorem.

Then:
 * $\alpha \left({A \cap B}\right) = \alpha \left({A}\right) \cap \alpha \left({B}\right)$