Quaternion Modulus of Product of Quaternions

Theorem
Let $\mathbf x, \mathbf y$ be quaternions.

Let $\size {\mathbf x}$ be the modulus of $\mathbf x$.

Then:
 * $\size {\mathbf {x y} } = \size {\mathbf x} \size {\mathbf y}$

Proof
Let $\mathbf x, \mathbf y$ be in their matrix form.

Then: