Coset Product is Well-Defined

Theorem
Let $\left({G, \circ}\right)$ be a group.

Let $N$ be a normal subgroup of $G$.

Let $a, b \in G$.

Then the coset product:
 * $\left({a N}\right) \left({b N}\right) = \left({a b}\right) N$

or alternatively, defined as:
 * $\left({a N}\right) \left({b N}\right) = \left\{{x y: x \in a N, y \in b N}\right\}$

is well-defined.

That is, the congruence modulo a subgroup is compatible with the group product.