Stokes Theorem for Oriented Sub-Rectangles in Euclidean Space

Let $n,k\in\mathbb{N}$ with $1\le k\le n$, let $(S,\varepsilon)$ be an oriented $k$-sub-rectangle of $\mathbb{R}^n$, let $U\subseteq \mathbb{R}^n$ be open with $S\subseteq U$, and let $\omega$ be a $C^1$ differential $(k-1)$-form on $U$. Then

$$\int_{(S,\varepsilon)} d\omega = \sum_{(T,\eta)\in \partial(S,\varepsilon)} \int_{(T,\eta)} \omega,$$

where $d\omega$ is the exterior derivative from Exterior Derivative of a C^1 Differential Form on a Euclidean Open Set, the left-hand side and each term on the right are integrals in the sense of Integral of a Differential Form over an Oriented k-Sub-Rectangle in Euclidean Space, and the boundary collection is the one from Boundary of an Oriented k-Sub-Rectangle in Euclidean Space.