Element of Group is in Unique Coset of Subgroup/Left

Theorem
Let $G$ be a group.

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

Let $x \in G$.

There exists a exactly one left coset of $H$ containing $x$, that is: $x H$

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

Also see

 * Element of Group in Unique Right Coset of Subgroup