Brahmagupta-Fibonacci Identity/Extension

Extension to Brahmagupta-Fibonacci Identity
Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be integers.

Then:
 * $\displaystyle \prod_{j \mathop = 1}^n \left({ {a_j}^2 + {b_j}^2}\right) = c^2 + d^2$

where $c, d \in \Z$.

That is: the product of any number of sums of two squares is also a sum of two squares.