holonomic ${\mathcal{D}}_{n-1}$-module, with no further conditions on $M$. After some reductions, it suffices to check that the *cokernel* of $x_n$ is holonomic, which is Proposition \[bjorkxn\]. These reductions are made possible by the following key lemma, which states that $R$-linear dependence relations among elements of $M_*$ hold homogeneously in $x_n$. (We write $\partial$ for $\partial_n$; this shorthand will be used throughout the rest of the paper.)

[@kernel Lemme 1]\[kerlemma\] Let $M$ be any ${\mathcal{D}}$-module, and let $M_* = \ker(\partial: M \rightarrow M)$. Suppose that $m_1, \ldots, m_l \in M_*$ are such that $f_1m_1 + \cdots + f_lm_l = 0$ for some $f_1, \ldots, f_l \in R$. Then $f_{1,j}m_1 + \cdots + f_{l,j}m_l = 0$ for every $j \geq 0$, where $f_{i,j} \in R_{n-1}$ denotes the coefficient of $x_n^j$ in $f_i$.

We first assume that the statement holds for $j = 0$ and prove that it follows for all $j > 0$. Note that for any $j \geq 0$ and any $f \in R$, we have $f_j = \frac{1}{j!}(\partial^j f)_0$, where the subscript $0$ denotes the constant term with respect to $x_n$. If $m_1, \ldots, m_l \in M_*$ and $f_1m_1 + \cdots + f_lm_l = 0$, then $\partial^j(f_1m_1 + \cdots + 