Example

Tag 0586, permanent. Cite it as [Bourbaki, Tag 0586].

Algebra, chapter I, § 3. Actions

(1) In the set $\mathscr{P}(E)$ of subsets of a set E, each of the internal laws $\cap$ and $\cup$ is distributive with respect to itself and the other. This follows from formulae of the form

A \cap (B \cup C) = (A \cap B) \cup (A \cap C)

A \cup (B \cap C) = (A \cup B) \cap (A \cup C).

(2) In $\mathbf{Z}$ (and more generally, in any totally ordered set) each of laws sup and inf is distributive with respect to the other and with respect to itself.

(3) In $\mathbf{Z}$ (and more generally in any ring) multiplication is distributive with respect to addition.

(4) In $\mathbf{N}$ addition and multiplication are distributive with respect to the laws sup and inf.

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

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