Definition 3

Tag 08JV, permanent. Cite it as [Bourbaki, Tag 08JV].

Integration, chapter II, § 1. Riesz spaces and fully lattice-ordered spaces

Let $E$ be an ordered vector space, $V$ and $W$ two supplementary linear subspaces of $E$ . One says that $E$ is the ordered direct sum of $V$ and $W$ if the canonical mapping $(x, y) \mapsto x + y$ of the ordered vector space $V \times W$ onto the ordered vector space $E$ is an isomorphism.

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

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