Proposition 8
Tag 06TY, permanent. Cite it as [Bourbaki, Tag 06TY].
Algebra, chapter V, § 15. Separable extensions
— Let L be an extension of a field K. If L is separable over K, every subextension of L is separable over K. Conversely, if every finitely generated subextension of L is separable over K, then L is separable over K.
This follows at once from Prop. 3, a) and b) (V, p. 119).
Thus separability may be called a property of « finite character ».
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.