Sum of Quaternion Conjugates

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

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

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

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

Then: