Subset Product of Subgroups/Necessary Condition

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

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

Let $H \circ K$ be a subgroup of $G$.

Then $H$ and $K$ are permutable.

That is: $H \circ K$ is a subgroup of $G$ iff:
 * $H \circ K = K \circ H$

where $H \circ K$ denotes subset product.

Proof
Suppose $H \circ K$ is a subgroup of $G$.

Then: