Brahmagupta-Fibonacci Identity/Extension/General

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

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

where $c, d \in \Z$.

That is: the set of all integers of the form $a^2 + m b^2$ is closed under multiplication.

Proof
Follows by induction from the general form of the Brahmagupta-Fibonacci Identity.