Corollary 3

Suppose that the modules $M/(x_1M+\cdots+x_{i-1}M)$ are separated for the $\mathfrak{x}$-adic topology $(1\leq i\leq n)$ ; let p be an integer between $1$ and $n$. In order that the sequence $\mathfrak{x}$ be completely secant for $M$, it is necessary and sufficient that the sequence $(x_1,\ldots,x_p)$ be completely secant for $M$ and that the sequence $(x_{p+1},\ldots,x_n)$ be completely secant for $M/(x_1M+\cdots+x_pM)$.

Indeed the corollary is evident if in the statement one replaces “completely secant sequences” by “regular sequences”; but the two notions coincide here by the theorem.