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 $\cmod z$ be the quaternion modulus of $z$.

Then:
 * $\cmod {\overline z} = \cmod z$