We argue by induction. The $N=2$ case is implied by Lemma 3.25. In general, we view $\sum_{1}^{N}|z_{i}|$ as a function of the imaginary parts of $z_{1},\ldots z_{N}$ subject to the constraints. Clearly this function achieves its minimum for some $(z_{i})$. If $z_{1}=a_{1}$, then we can conclude by induction. Otherwise $\text{Im}z_{1}>\text{Im}a_{1}$. If $\arg z_{2}<\arg z_{1}$, then we can fix $z_{1}+z_{2}$ and decrease $\sum_{1}^{2}|z_{i}|$ by Lemma 3.25, which would contradict minimality. Proceding with this argument, we are forced to have

$$\arg z_{1}\leq\arg z_{2}\leq\ldots\leq\arg z_{N},$$

whence

$$\arg\sum_{1}^{N}z_{i}\geq\arg z_{1}>\arg a_{1}>\arg\sum_{1}^{N}a_{i}$$

which contradicts $\sum_{1}^{N}z_{i}=\sum_{1}^{N}a_{i}$. ∎