Corollary 2

Let A be a local ring, J an ideal of finite type of A distinct from A, M a nonzero A-module of finite type. Put $r = [J/m_AJ : \kappa_A]$. One has $\operatorname{prof}_A(J; M) \leq r$ ; there is equality if and only if J is generated by a family completely secant for M. In this case, in order that a generating family of J be completely secant, it is necessary and sufficient that it have r elements.

By Nakayama's lemma, one has $JM \neq M$, and r is the minimal number of generators of J; Corollary 2 therefore follows from Corollary 1.