Corollary 1

Tag 04EU, permanent. Cite it as [Bourbaki, Tag 04EU].

Commutative Algebra, chapter III, § 2. General results on filtered rings and modules

Under the conditions of Proposition 16, if $A$ is complete, so is $E$ .

As the canonical mapping $A \to \hat{A}$ is then surjective (no. 6, Proposition 5), $\hat{E} = i(E)$ by (25) and the conclusion follows by Proposition 5 of no. 6.

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

Cited by

Corollary 4, § 2

Cites

Proposition 16
Proposition 5
no. 6, Proposition 5