Proposition 5

— If $(x_i)$ is a finite family of elements of a filtered group G, then there exists $z \in G$ such that $x_i + z$ is positive for each i.

If $x_i = u_i - v_i$, with $u_i$ and $v_i$ positive, it is enough to take $z$ to be the sum of the family $(v_i)$.