Subset Products of Normal Subgroup with Normal Subgroup of Subgroup

Theorem
Let $G$ be a group.

Let:
 * $(1): \quad H$ be a subgroup of $G$
 * $(2): \quad K$ be a normal subgroup of $H$
 * $(3): \quad N$ be a normal subgroup of $G$

Then:
 * $N K \lhd N H$

where:
 * $N K$ and $N H$ denote subset product
 * $\lhd$ denotes the relation of being a normal subgroup.

Proof
Consider arbitrary $x_n \in N, x_h \in H$.

Thus:
 * $x_n x_h \in N H$

We aim to show that:
 * $x_n x_h N K \paren {x_n x_h}^{-1} \subseteq N K$

thus demonstrating $N K \lhd N H$ by the Normal Subgroup Test.

We have: