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

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

Then:
 * $\left\lvert{\alpha + \beta}\right\rvert^2 + \left\lvert{\alpha - \beta}\right\rvert^2 = 2 \left\lvert{\alpha}\right\rvert^2 + 2 \left\lvert{\beta}\right\rvert^2$

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

Then: