Proposition 7
Tag 04NE, permanent. Cite it as [Bourbaki, Tag 04NE].
Commutative Algebra, chapter V, § 1. Notion of an integral element
Let A be a ring and R an A-algebra. The integral closure A' of A in R is a subring integrally closed in R.
The integral closure of A' in R is integral over A by no. 1, Proposition 6; it is therefore equal to A'.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.