Equivalence of Definitions of Complement of Subgroup

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

Let $H$ and $K$ be subgroups.

1 implies 2
Let $G = HK$.

Then $HK$ is a group.

By Subset Product of Subgroups, $HK=KH$.

Thus $KH = G$.

1 implies 2
Let $G = KH$.

Then $KH$ is a group.

By Subset Product of Subgroups, $HK=KH$.

Thus $HK = G$.