Exercise 10, § 1

Let $A$ be a noetherian ring, $M$ a finitely generated $A$-module, $J$ an ideal of $A$, $(x_1,\ldots,x_r)$ an $M$-regular sequence of elements of $J$, with $r < \mathrm{prof}_A(J;M)$. Let $q_1,\ldots,q_n$ be ideals of $A$ not containing $J$, all prime except at most two, and $J_0$ a subset of $J$, stable under addition and multiplication and generating the ideal $J$. Prove that there exists an element $x$ of $J_0$ belonging to none of the $q_i$ such that the sequence $(x_1,\ldots,x_r,x)$ is $M$-regular (use cor. 2 of prop. 2 of II, § 1, No. 1).