Corollary 2
Tag 06TV, permanent. Cite it as [Bourbaki, Tag 06TV].
Algebra, chapter V, § 15. Separable extensions
— Let A and B be two commutative K-algebras. If A and B are separable, the same is true of A ⊗_K B.
Let L be an extension of K. The ring L ⊗_K A is reduced because A is separable; now Prop. 5 shows the ring (L ⊗_K A) ⊗_K B (isomorphic to L ⊗_K (A ⊗_K B)) to be reduced, whence the Corollary.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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