In view of Definition 7.1 both formulae in the above statement are equivalent and so it is enough to prove the second one. In case $u=0$, we have $\Delta(u)=\{\Delta\}$ and formula (7.4) holds because

$$\int_{\Delta}f^{(n)}(\langle 0,x\rangle)\,\text{\rm d}\operatorname{vol}_{n}=\operatorname{vol}(\Delta)f^{(n)}(0)=\sum_{k\geq 0}C_{k}(\Delta,0,\Delta)f^{(k)}(0),$$

We prove (7.4) by induction on the dimension $n$. In case $n=0$, we have $u=0$ and so the verification reduces to the above one. Hence, we assume $n\geq 1$ and $u\neq 0$. For short, we write $\,\text{\rm d}x=\,\text{\rm d}x_{1}\wedge\dots\wedge\,\text{\rm d}x_{n}$. Choose any vector $v\in\mathbb{R}^{n}$ of norm $1$ and such that $\langle u,v\rangle\not=0$. Performing an orientation-preserving orthonormal change of variables, we may assume $v=(1,0,\dots,0)$. We have

$$f^{(n)}(\langle u,x\rangle)\,\text{\rm d}x=\frac{1}{\langle u,v\rangle}\,\text{\rm d}\left(f^{(n-1)}(\langle u,x\rangle)\,\text{\rm d}x_{2}\wedge\dots\wedge\,\text{\rm d}x_{n}\right).$$

With Stokes’ theorem, we obtain

(7.5)		$\displaystyle\int_{\Delta}f^{(n)}(\langle u,x\rangle)\,\text{\rm d}\operatorname{vol}_{n}$	$\displaystyle=\int_{\Delta}f^{(n)}(\langle u,x\rangle)\,\text{\rm d}x$
		$\displaystyle=\frac{1}{\langle u,v\rangle}\sum_{F}\int_{F}f^{(n-1)}(\langle u,x\rangle)\,\text{\rm d}x_{2}\wedge\dots\wedge\,\text{\rm d}x_{n}.$

where the sum is over the facets $F$ of $\Delta$, and we equip each facet with the induced orientation.

For each facet $F$ of $\Delta$, we let $\iota_{u_{F}}(\,\text{\rm d}x)$ be the differential form of order $n-1$ obtained by contracting $\,\text{\rm d}x$ with the vector $u_{F}$. The form $\,\text{\rm d}x_{2}\wedge\dots\wedge\,\text{\rm d}x_{n}$ is invariant under translations and its restriction to the linear hyperplane $L_{F}$ coincides with $\langle u_{F},v\rangle\iota_{u_{F}}(\,\text{\rm d}x)$. Therefore,

$$\int_{F}f^{(n-1)}(\langle u,x\rangle)\,\text{\rm d}x_{2}\wedge\dots\wedge\,\text{\rm d}x_{n}=\langle u_{F},v\rangle\int_{F-m_{F}}f^{(n-1)}(\langle u,x+m_{F}\rangle)\iota_{u_{F}}(\,\text{\rm d}x).$$

Let $\operatorname{vol}_{n-1}$ denote the Lebesgue measure on $L_{F}$. We can verify that $\operatorname{vol}_{n-1}$ coincides with the measure induced by integration of $-\iota_{u_{F}}(\,\text{\rm d}x)$ along $L_{F}$. Let $g\colon\mathbb{R}\to\mathbb{R}$ be the function defined as $g(z)=f(z+\langle u,m_{F}\rangle)$. Then $f^{(n-1)}(\langle u,x+m_{F}\rangle)=g^{(n-1)}(\langle\pi_{F}(u),x\rangle)$ for all $x\in L_{F}$. Hence,

$$\int_{F-m_{F}}f^{(n-1)}(\langle u,x+m_{F}\rangle)\iota_{u_{F}}(\,\text{\rm d}x)=-\int_{F-m_{F}}g^{(n-1)}(\langle\pi_{F}(u),x\rangle)\,\text{\rm d}\operatorname{vol}_{n-1}.$$

Applying the inductive hypothesis to $F$ and the function $g$ we obtain

	$\displaystyle\int_{F}g^{(n-1)}(\langle\pi_{F}(u),x\rangle)\,\text{\rm d}\operatorname{vol}_{n-1}$	$\displaystyle=\sum_{V^{\prime}\in F(\pi_{F}(u))}\sum_{k\geq 0}C_{k}(F,\pi_{F}(u),V^{\prime})g^{(k)}(\langle\pi_{F}(u),V^{\prime}\rangle)$
		$\displaystyle=\sum_{V^{\prime}\in F(\pi_{F}(u))}\sum_{k\geq 0}C_{k}(F,\pi_{F}(u),V^{\prime})f^{(k)}(\langle u,V^{\prime}\rangle).$

Each aggregate $V^{\prime}\in F(\pi_{F}(u))$ is contained in a unique $V\in\Delta(u)$ and it coincides with $V\cap F$. Therefore, we can transform the right-hand side of the last equality in

$$\sum_{V\in\Delta(u)}\sum_{k\geq 0}C_{k}(F,\pi_{F}(u),V\cap F)f^{(k)}(\langle u,V\rangle),$$

where, for simplicity, we have set $C_{k}(F,\pi_{F}(u),V\cap F)=0$ whenever $V\cap F=\emptyset$. Plugging the resulting expression into (7.5) and exchanging the summations on $V$ and $F$, we obtain that $\int_{\Delta}f^{(n)}(\langle x,u\rangle)\,\text{\rm d}\operatorname{vol}_{n}$ is equal to

(7.6)

$$\sum_{V\in\Delta(u)}\sum_{k\geq 0}\bigg(-\sum_{F}\frac{\langle u_{F},v\rangle}{\langle u,v\rangle}C_{k}(F,\pi_{F}(u),V\cap F)f^{(k)}(\langle u,V\rangle)\bigg).$$

Specialising this identity to $v=u$, we readily derive formula (7.4) from Definition 7.1 of the coefficients $C_{k}(\Delta,u,V)$.

For the last statement, observe that the values $f^{(k)}(\langle u,V\rangle)$ can be arbitrarily chosen. Hence, the coefficients $C_{k}(\Delta,u,V)$ are uniquely determined from the linear system obtained from the identity (7.4) for enough functions $f$. ∎