Normal Subgroup of Subset Product of Subgroups

Theorem
Let $G$ be a group whose identity is $e$.

Let:
 * $H$ be a subgroup of $G$
 * $N$ be a normal subgroup of $G$.

Then:
 * $N \lhd N H$

where:
 * $\lhd$ denotes normal subgroup
 * $N H$ denotes subset product.

Proof
From Subset Product with Normal Subgroup is Subgroup we have that $N H = H N$ is a subgroup of $G$.

Now to show that $N$ is a normal subgroup of $N H$