This question regards Lemma 2.29 on page 27 of these Measure Theory notes:
https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch2.pdf
I am trying to use the method presented in these notes to derive the invariance of the Lebesgue measure under orthogonal linear operators. All steps in the proof are quite clear except for this lemma, which is handwaved in the following manner.
The text defines "oblique rectangles" $\tilde{R}$ as parallelepipeds with orthonormal axes, and defines their volume $v(\tilde{R})$ in the obvious manner: as the product of the side lengths. The bulk of the next few pages are designed to prove that the volume of an oblique rectangle is equal to its Lebesgue measure.
The Lemma I am struggling with is the following. Note that the text uses "parallel rectangles" to refer to typical boxes in $\mathbf{R}^n$, and for a parallel rectangle $R$, $v(R)$ refers to its volume, which we already know is equal to its outer measure and Lebesgue measure.
Lemma 2.29. If an oblique rectangle $\tilde{R}$ contains a finite almost disjoint collection of parallel rectangles $\{R_1, R_2 \dots, R_N\}$ then
\begin{equation*} \sum_{i=1}^N v(R_i) \leq v(\tilde{R}) \end{equation*}
The only justification offered is: "This result is geometrically obvious, but a formal proof seems to require a fuller discussion of the volume function on elementary geometrical sets, which is included in the theory of valuations in convex geometry. We omit the details."
I would like to know if there is a way to prove this fact without developing too much extra machinery. I know the final result of invariance of Lebesgue measure through orthogonal operators is discussed elsewhere in terms of two common proofs: one, from e.g. Rudin, which uses general Borel measure theory, and one, from e.g. Frank Jones, which uses decomposition of matrices into elementary matrices. I am asking specifically about this alternate (some may say more ad. hoc) method, because I believe it may require less external machinery to be developed. I know that the question:
How do I prove that $m(TE)=|\det(T)|m(E)$?
makes reference to the same lemma I quoted here, but I do not believe this question is a duplicate. The linked page asks a more general question about images of sets through linear operators, and the only answer provides a standard proof using elementary matrices, without proving the lemma along the way. In my case, I am specifically interested in proving the lemma without using one of the two aforementioned methods.
For reference to where I am in my knowledge of these fields, the Lebesgue measure theory I know is from Tao's "Analysis II" Chapters 7 and 8, and I'm trying to develop rigorous analogs to the theorems in the latter section of Axler's "Linear Algebra Done Right" Chapter 10.