Brahmagupta-Fibonacci Identity/Extension

Corollary 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.

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

where $c, d \in \Z$.

That is: the set of all numbers of the form $x^2 + n y^2$ is closed under multiplication.

Proof
Follows by induction from the Brahmagupta-Fibonacci Identity.