Corollary 5

— For any integer $n \geq 0$ we have $(nx)^+ = nx^+$ and $(nx)^- = nx^-$; for all $n \in \mathbf{Z}$ we have $|nx| = |n| \cdot |x|$. Indeed $nx = nx^+ - nx^-$; since $x^+$ and x are coprime, so are $nx^+$ and $nx^-$ if $n \geq 0$ (Cor. 4); the first assertion follows by Prop. 9 b) of VI, p. 12. The second follows from the first by Prop. 9 d) in the case $n \geq 0$; the case $n < 0$ follows from this as a result of the relation $|-x| = |x|$.