Equivalent Statements for Congruence Modulo Subgroup

Left Congruence
Let $G$ be a group, and let $H$ be a subgroup of $G$.

Let $x \ \equiv^l \ y \ \left({\bmod H}\right)$ denote that $x$ is left congruent modulo $H$ to $y$.

Then the following statements are equivalent:

Right Congruence
Let $G$ be a group, and let $H$ be a subgroup of $G$.

Let $x \ \equiv^r \ y \ \left({\bmod H}\right)$ denote that $x$ is right congruent modulo $H$ to $y$.

Then the following statements are equivalent:

Proof
Each statement follows directly from the previous one, by definition of Congruence Modulo a Subgroup.