Remark 1

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

Commutative Algebra, chapter VIII, § 2. Dimension des algèbres

If the rings $A$ and $B$ are noetherian, we shall see below (§ 3, no 4, cor. 2 to prop. 7) that one has the inequality $\dim(B) \leq \dim(A) + n$ , stronger than that of prop. 3.

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

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