Right Cosets are Equal iff Element in Other Right Coset

Theorem
Let $G$ be a group.

Let $H$ be a subgroup of $G$.

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

Then:
 * $H x = H y \iff x \in H y$

Also see

 * Left Cosets are Equal iff Element in Other Left Coset