Right Cosets are Equal iff Product with Inverse in Subgroup

Theorem
Let $G$ be a group and let $H$ be a subgroup of $G$.

Let $x, y \in G$. Let $H x$ denote the right coset of $H$ by $x$.

Then:
 * $H x = H y \iff x y^{-1} \in H$

Also see

 * Left Cosets are Equal iff Product with Inverse in Subgroup