Element in Coset 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:
 * $y H$ denote the left coset of $H$ by $y$;
 * $H y$ denote the right coset of $H$ by $y$.

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

Proof

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


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