Combination Theorem for Complex Derivatives/Sum Rule

Theorem
Let $D$ be an open subset of the set of complex numbers $\C$.

Let $f, g: D \to \C$ be complex-differentiable functions on $D$

Then $f + g$ is complex-differentiable in $D$, and its derivative $\paren {f + g}'$ is defined by:
 * $\map {\paren {f + g}'} z = \map {f'} z + \map {g'} z$

for all $z \in D$.