Intersection of Left Cosets of Subgroups is Left Coset of Intersection

Theorem
Let $G$ be a group.

Let $H, K \le G$ be subgroups of $G$.

Let $a, b \in G$.

Let:
 * $a H \cap b K \ne \O$

where $a H$ denotes the left coset of $H$ by $a$.

Then $a H \cap b K$ is a left coset of $H \cap K$.

Proof
Let $x \in a H \cap b K$.

Then:

and similarly:

Hence:

Hence the result by definition of left coset.