Corollary 4

Tag 06AD, permanent. Cite it as [Bourbaki, Tag 06AD].

— Let $f, g, h \in K[X]$ . If f divides gh and is prime to g, then f divides h. For f divides gh and fh, hence f divides every gcd of gh and fh, in particular h (Cor. 2, (i)).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.