Element of Group is in Unique Coset of Subgroup

Theorem
Let $G$ be a group, and let $H \le G$.

Then:
 * For each $x \in G$ there exists a unique left coset of $H$ containing $x$, that is: $x H$
 * For each $x \in G$ there exists a unique right coset of $H$ containing $x$, that is: $H x$.

Proof
Follows directly from:
 * Congruence Modulo Subgroup is Equivalence Relation
 * Element in its own Equivalence Class.