Brahmagupta-Fibonacci Identity/Proof 4

Proof
Let $z_1 = a + b i, z_2 = c + d i$ be complex numbers.

Let $\left\lvert{z}\right\rvert$ denote the complex modulus of a given complex number $z \in \C$.

By definition of complex multiplication:


 * $(1): \quad z_1 z_2 = \left({a c - b d}\right) + \left({a d + b c}\right) i$

Then we have: