Proposition 6

— Let $(x_i)_{i \in A}, (y_\beta)_{\beta \in B}$ be two families of elements of an ordered group G, each having a supremum. Then the family $(x_\alpha + y_\beta)_{(\alpha, \beta) \in A \times B}$ has a supremum, and

Indeed, from $x_\alpha + y_\beta \leq z$ for all $\alpha$ and $\beta$, we deduce $\sup(x_i) + \sup(y_\beta) \leq z$ for all $\beta$, and hence $\sup(x_i) + \sup(y_\beta) \leq z$.

1 When A is the ring of integers (resp. the polynomial ring in one indeterminate with coefficients in a field), these definitions coincide with those of I, p. 112 (resp. IV, p. 12, Def. 1).