Proposition 2

Every family $(x_i)$ of real numbers $\geq 0$ is summable in $\overline{\mathbf{R}}$.

For the mapping $H \to s_H$ of the directed set $\mathfrak{F}(I)$ into $\overline{\mathbf{R}}$ is increasing; hence (§ 5, no. 2, Theorem 2) has a limit.

The same reasoning shows that every family of real numbers $\leq 0$ is summable in $\overline{\mathbf{R}}$.

Similarly, multiplication is an associative and commutative law of composition on each of the intervals $[0, 1]$ and $[1, + \infty]$ of $\overline{\mathbf{R}}$, and therefore the notion of a multipliable family is defined on each of these intervals.