Sum of Quaternion Conjugates

Theorem
Let $\mathbf x,\mathbf y\in\mathbb H$ be quaternion.

Let $\overline{\mathbf x}$ be the conjugate of $\mathbf x$.

Then:
 * $\overline{\mathbf x + \mathbf y} = \overline{\mathbf x} + \overline{\mathbf y}$

Proof
Let $\mathbf x = a\mathbf 1+b\mathbf i+c\mathbf j+d\mathbf k$ and $\mathbf y = e\mathbf 1+f\mathbf i+g\mathbf j+h\mathbf k$.

Then: