Intersection with Normal Subgroup is Normal/Examples/Subset Product of Normal Subgroup with Intersection

Theorem
Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Let $H_1, H_2$ be subgroups of $G$.

Let:
 * $N_1 \lhd H_1$
 * $N_2 \lhd H_2$

where $\lhd$ denotes the relation of being a normal subgroup.

Then:

where $N_1 \circ H_1$ denotes the subset product of $N_1$ and $H_1$.