Left Cosets are Equal iff Product with Inverse in Subgroup

Theorem
Let $G$ be a group.

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

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

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

Also see

 * Right Cosets are Equal iff Product with Inverse in Subgroup