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.