Element in Coset iff Product with Inverse in Subgroup

Theorem
Let $\left({G, \circ}\right)$ be a group and let $H$ be a subgroup of $G$.

Let $x, y \in G$.

Let:
 * $y \circ H$ denote the left coset of $H$ by $y$
 * $H \circ y$ denote the right coset of $H$ by $y$.

Then: