Brahmagupta-Fibonacci Identity/Extension/Proof 3

Proof
Let $z_1 = a_1 + i b_1, z_2 = a_2 + i b_2, \ldots, z_n = a_n + i b_n$.

Let $c + i d = z_1 z_2 \cdots z_n$.

Then:

As $a_1, a_2, \dotsc, a_n$ and $b_1, b_2, \dotsc, b_n$ are positive integers, then so are $c$ and $d$.

Thus: