Subgroup Subset of Subgroup Product

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

Let $H$ and $K$ be subgroups of $G$.

Then:
 * $H \subseteq H \circ K \supseteq K$

where $H \circ K$ denotes the subset product of $H$ and $K$.

Proof
By definition of subset product:
 * $H \circ K = \left\{{h \circ k: h \in H, k \in K}\right\}$

So:

and: