Corollary

Tag 03JP, permanent. Cite it as [Bourbaki, Tag 03JP].

Theory of Sets, chapter III, § 1. Order relations. Ordered sets

Let $(x_{\lambda\mu})_{(\lambda,\ \mu) \in L \times M}$ be a "double" family of elements of an ordered set $\mathrm{E}$ such that for each $\mu \in \mathrm{M}$ the family $(x_{\lambda\mu})_{\lambda \in L}$ has a least upper bound in $\mathrm{E}$ . For the family $(x_{\lambda\mu})_{(\lambda,\ \mu) \in L \times M}$ to have a least upper bound in $\mathrm{E}$ , it is necessary and sufficient that the family $\left( \sup\limits_{\lambda \in L} x_{\lambda\mu} \right)_{\mu \in M}$ should have a least upper bound in E, and then we have

(2) \qquad \sup_{(\lambda,\,\mu)\in L\times M} x_{\lambda\mu} = \sup_{\mu\in M}\left(\sup_{\lambda\in L} x_{\lambda\mu}\right).

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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