Corollary

Every sequence of elements of $m_A$ which is completely secant for M (A, X, p. 157, Definition 2) is secant for M. Let $(x_1, ..., x_r)$ be a sequence of elements of $m_A$ which is completely secant for M. Put $M_0 = M$ and by induction $M_i = M_{i-1}/x_i M_{i-1}$ for $1 \leq i \leq r$. By Corollary 1 of A, X, p. 160, the homothety of ratio $x_i$ in $M_{i-1}$ is injective, whence $\dim_A(M_i) = \dim_A(M_{i-1}) - 1$ (for $1 \leq i \leq r$) (Proposition 3). Therefore one has

hence the sequence $(x_1, ..., x_r)$ is secant for M.