Sum of Squares of Complex Moduli of Sum and Differences of Complex Numbers

Theorem
Let $\alpha, \beta \in \C$ be complex numbers.

Then:
 * $\cmod {\alpha + \beta}^2 + \cmod {\alpha - \beta}^2 = 2 \cmod \alpha^2 + 2 \cmod \beta^2$

Proof
Let:
 * $\alpha = x_1 + i y_1$
 * $\beta = x_2 + i y_2$

Then: