If $x \in \mathbb{R}^K$, then \begin{equation} \left( \prod_{n = 1}^N f_n \right)(-x) = \prod_{n = 1}^N f_n(-x) = \prod_{n = 1}^N f_n(x) = \left( \prod_{n = 1}^N f_n \right)(x) \end{equation} $\square$