Intersection of Quotient Groups

Theorem
Let $$N \triangleleft G$$, 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 $$\alpha \left({A \cap B}\right) = \alpha \left({A}\right) \cap \alpha \left({B}\right)$$.

Proof
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