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:
 * $N H = H N$ is a subgroup of $G$.

By definition of subset product all elements of $H N$ can be written in the form:
 * $h n \in H N$

where $h \in H, n \in N$.

Let $h n \in H N$.

Let $n_1 \in N$.

From Inverse of Group Product:
 * $\paren {h n} n_1 \paren {h n}^{-1} = h n n_1 n^{-1} h^{-1}$

We have that:
 * $n n_1 n \in N$
 * $h, h^{-1} \in G$.

Then, since $N$ is a normal subgroup of $G$:
 * $\paren {h n} n_1 \paren {n^{-1} h^{-1} } \in N$

Thus:
 * $N \lhd N H$