Let $R$ be the division ring over which $\prod_{n = 1}^N V_n$ and $W$ are vector spaces and let $0_R$ be the additive identity in $R$. Since $0_n = 0_R 0_n$, since $0_R w = 0_W$ for every $w \in W$, and since $f$ is homogeneous in each argument, then \begin{equation} \begin{split} f(x) = f(x_1, \ldots, x_N) & = f(x_1, \ldots, x_n, \ldots, x_N) \\ & = f(x_1, \ldots, 0_n, \ldots, x_N) \\ & = f(x_1, \ldots, 0_R 0_n, \ldots, x_N) = 0_R f(x_1, \ldots, 0_n, \ldots, x_N) = 0_W \end{split} \end{equation} $\square$