Proposition 7

Let $\mathbf{f}$ be a regulated vector function on a compact interval $I = [a, b]$, with values in a complete normed space $E$, and let $g$ be a real regulated function on $I$, such that $g(x) \geq 0$ at the points where $g$ is continuous; in these circumstances

(15)

In particular,

(16)