Corollary 7
Tag 05PH, permanent. Cite it as [Bourbaki, Tag 05PH].
Algebra, chapter II, § 3. Tensor products
Let C be a commutative ring. If E and F are two projective C-modules, is a projective C-module.
This follows immediately from Corollary 5 and the fact that the tensor product of two free C-modules is a free C-module (Corollary 2).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Nothing.