Equivalent Statements for Congruence Modulo Subgroup/Left

Theorem
Let $G$ be a group.

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

Let $x \equiv^l y \pmod H$ denote that $x$ is left 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.