Corollary 1

Tag 0D16, permanent. Cite it as [Bourbaki, Tag 0D16].

Commutative Algebra, chapter X, § 1. Profondeur

One has the inequality

\dim(M_q) - prof_{A_q}(M_q) \geq \dim(M_p) - prof_{A_p}(M_p) \geq 0 .

This follows from prop. 13 and the inequality $\dim(M_q) \geq \dim(M_p) + \dim(A_q/pA_q)$ (VIII, § 1, No. 4, prop. 9, a) and No. 3, prop. 7, b)).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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