Corollary 2
Tag 0737, permanent. Cite it as [Bourbaki, Tag 0737].
Algebra, chapter VII, § 3. Free modules over a principal ideal domain
— Every submodule of a free module over a principal ideal domain is free. This follows immediately from Th. 1, since every ideal of a principal ideal domain is free.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.