Subset Product with Normal Subgroup as Generator

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:
 * $N \lhd \gen {N, H} = N H = H N \le G$

where:
 * $\le$ denotes subgroup
 * $\lhd$ denotes normal subgroup
 * $\gen {N, H}$ denotes a subgroup generator
 * $N H$ denotes subset product.

Proof
From Subset Product is Subset of Generator:
 * $N H \subseteq \gen {N, H}$

From Subset Product with Normal Subgroup is Subgroup:
 * $N H = H N \le G$

Then by the definition of a subgroup generator, $\gen {N, H}$ is the smallest subgroup containing $N H$ and so:
 * $\gen {N, H} = N H = H N \le G$

From Normal Subgroup of Subset Product of Subgroups we have that:
 * $N \lhd N H$

Hence the result.