Quaternion Modulus of Product of Quaternions

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

Let $\left\vert{\mathbf x}\right\vert$ be the modulus of $\mathbf x$.

Then:
 * $\left\vert{\mathbf x \mathbf y}\right\vert = \left\vert{\mathbf x}\right\vert \left\vert{\mathbf y}\right\vert$

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

Then: