Element in Left Coset iff Product with Inverse in Subgroup

Theorem
Let $\struct {G, \circ}$ be a group.

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

Let $x, y \in G$.

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

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

Also see

 * Element in Right Coset iff Product with Inverse in Subgroup