Corollary 1
Tag 05N6, permanent. Cite it as [Bourbaki, Tag 05N6].
Algebra, chapter II, § 2. Modules of linear mappings. Duality
For an A-module to be projective and finitely generated, it is necessary and sufficient that it be a direct factor of a free A-module with a finite basis.
The condition is obviously sufficient; conversely, a finitely generated projective module E is isomorphic to a quotient of a free module F with a finite basis (§ 1, no. 11) and E is isomorphic to a direct factor of F by virtue of Proposition 4 (d).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.