Let $f : \prod_{n = 1}^N V_n \to W$ be a D1423: Multilinear map such that

(i)	$0_1, \ldots, 0_N, 0_W$ are each the D737: Zero vector in $V_1, \ldots, V_N, W$, respectively

Let $x = (x_1, \ldots, x_N) \in \prod_{n = 1}^N V_n$ be a D1129: Vector in $\prod_{n = 1}^N V_n$ such that

(i)	\begin{equation} \exists \, n \in 1, \ldots, N : x_n = 0_n \end{equation}

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} $\square$