Equivalence of Definitions of Complement of Subgroup

Theorem
Let $G$ be a group with identity $e$.

Let $H$ and $K$ be subgroups.

Definition $1$ implies Definition $2$
Let $G = H K$.

Then $H K$ is a group.

By Subset Product of Subgroups:
 * $H K = K H$

Thus $K H = G$.

Definition $2$ implies Definition $1$
Let $G = K H$.

Then $K H$ is a group.

By Subset Product of Subgroups:
 * $H K = K H$

Thus $H K = G$.