Quaternion Modulus of Conjugate

Theorem
Let $z = a\mathbf 1+b\mathbf i+c\mathbf j+d\mathbf k$ be a quaternion.

Let $\overline z$ be the conjugate of $z$.

Let $\vert z\vert$ be the quaternion modulus of $z$.

Then:
 * $\vert \overline z\vert = \vert z\vert$