Corollary 2
Tag 04EQ, permanent. Cite it as [Bourbaki, Tag 04EQ].
Commutative Algebra, chapter III, § 2. General results on filtered rings and modules
Under the hypotheses of Corollary 1, for to be Noetherian, it is necessary and sufficient that , be so.
The condition is necessary since , is isomorphic to a quotient of ; it is sufficient by virtue of no. 10, Corollary 5 to Theorem 2.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.