Corollary 1

*If A is noetherian, if the A-module M is of finite type and if the $x_i$ belong to the radical of A, the conditions (i) to (v) of the theorem are equivalent. Indeed, on each of the modules $M/(x_1 M + \cdots + x_{i-1} M)$ the $x$-adic topology is separated (AC III, § 3, No. 3, prop. 6).