Let $z, w \in \mathbb{C}$ each be a D1207: Complex number.

Denote $z = (a, b)$ and $w = (c, d)$. Using the definitions

(i)	D607: Complex addition operation
(ii)	D507: Complex conjugate

we have \begin{equation} \begin{split} \overline{z + w} = \overline{(a, b) + (c, d)} & = \overline{(a + c, b + d)} \\ & = (a + c, - b - d) \\ & = (a, - b) + (c, - d) \\ & = \overline{(a, b)} + \overline{(c, d)} \\ & = \overline{z} + \overline{w} \end{split} \end{equation} $\square$