Coset Equals Subgroup iff Element in Subgroup

Theorem
Let $G$ be a group and let $H$ be a subgroup of $G$.

Let $x, y \in G$.

Let:
 * $x H$ denote the left coset of $H$ by $x$
 * $H x$ denote the right coset of $H$ by $x$.

Then:
 * $x H = H \iff x \in H$
 * $H x = H \iff x \in H$

Proof

 * $x H = H \iff x \in H$:


 * $H x = H \iff x \in H$ is proved similarly: