Proposition 7

Tag 06WW, permanent. Cite it as [Bourbaki, Tag 06WW].

Algebra, chapter VI, § 1. Ordered groups. Divisibility

— *If x and y are two elements of an ordered group G, and if one of the elements $\inf(x, y)$ , $\sup(x, y)$ exists, then so does the other, and $x + y = \inf(x, y) - \sup(x, y)$ .

Indeed, according to the relations (1) and (2) (VI, p. 9) we have

\sup(a - x, a - y) = a + \sup(-x, -y) = a - \inf(x, y),

and it suffices to take $a = x + y$ .

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

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