Proof. We have

$$

\begin{aligned}

&\left|z_{1}-z_{2}\right|^{2}+\left|z_{1}+z_{2}\right|^{2} \\

=&\left(z_{1}-z_{2}\right) \overline{\left(z_{1}-z_{2}\right)}+\left(z_{1}+z_{2}\right) \overline{\left(z_{1}+z_{2}\right)} \\

=&\left(z_{1}-z_{2}\right)\left(\bar{z}_{1}-\bar{z}_{2}\right)+\left(z_{1}+z_{2}\right)\left(\bar{z}_{1}+\bar{z}_{2}\right) \\

=&\left(\left(z_{1} \bar{z}_{1}+z_{2} \bar{z}_{2}\right)-\left(z_{1} \bar{z}_{2}+z_{2} \bar{z}_{1}\right)\right)+\left(\left(z_{1} \bar{z}_{1}+z_{2} \bar{z}_{2}\right)+\left(z_{1} \bar{z}_{2}+z_{2} \bar{z}_{1}\right)\right) \\

=& 2\left(z_{1} \bar{z}_{1}+z_{2} \bar{z}_{2}\right)=2\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right)

\end{aligned}

$$