Let $I \subseteq \mathbb{R}^N$ be a Euclidean real interval and let $\lambda \in \mathbb{R}$. By definition, $I$ is a finite Cartesian product of some basic real intervals $I_1, \dots, I_N$. Denote the respective endpoints of these intervals by $a_1, b_1, \dots, a_N, b_N$.
Result
R2973: The four classes of real intervals under reflection shows, among other things, that a reflection of an interval preserves its length. Therefore, we may assume that $\lambda \geq 0$. Result
R2972: The four classes of real intervals are each closed under dilation shows that the dilated intervals $\lambda I_1, \dots, \lambda I_N$ have respective endpoints $\lambda a_1, \lambda b_1, \dots, \lambda a_N, \lambda b_N$. In the case that $\lambda$ equals zero, we may treat all intervals as closed, without this affecting the respective lengths, so that the respective endpoints stay well-defined.
Thus, applying
R2969: Euclidean real Cartesian product of scaled sets, one concludes
\begin{equation}
\begin{split}
\mathsf{Vol}(\lambda I) & = \mathsf{Vol} \Big( \lambda \prod_{n = 1}^N I_n \Big) \\
& = \mathsf{Vol} \Big( \prod_{n = 1}^N \lambda I_n \Big) \\
& = \prod_{n = 1}^N |\lambda b_n - \lambda a_n| \\
& = |\lambda|^N \prod_{n = 1}^N |b_n - a_n| \\
& = |\lambda|^N \mathsf{Vol} \Big( \prod_{n = 1}^N I_n \Big) \\
& = |\lambda|^N \mathsf{Vol} (I) \\
\end{split}
\end{equation}
$\square$
