Corollary 4
Tag 0048, permanent. Cite it as [Bourbaki, Tag 0048].
Algebra, VIII, p. 57, § 4. Semisimple Modules
Let M be a semisimple module. Then M is simple if and only if the endomorphism ring E of M is a field.
If M is simple, then E is a field by the corollary of Proposition 2 of VIII, p. 47.
If E is a field, then the module M is indecomposable (VIII, p. 31, Proposition 4, a)). Since it is moreover semisimple, it is simple.
Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.