Statement (1) follows easily from the definition of $C_{k}(\Delta,u,V)$. For statement (2), we use that, from (7.6), the integral formula in Proposition 7.3 also holds for the choice of coefficients

$$-\sum_{F}\frac{\langle u_{F},v\rangle}{\langle u,v\rangle}C_{k}(F,\pi_{F}(u),V\cap F)$$

for any vector $v$ of norm 1 such that $\langle u,v\rangle\neq 0$. But the coefficients satisfying that formula are unique. Hence, this choice necessarily coincides with $C_{k}(\Delta,u,V)$ for all such $v$. Hence,

$$\langle u,v\rangle C_{k}(\Delta,u,V)=-\sum_{F}\langle u_{F},v\rangle C_{k}(F,\pi_{F}(u),V\cap F)$$

and formula (7.9) follows. Statement (3) follows from Formula (7.4) applied to $\Delta$, $\Delta_{1}$ and $\Delta_{2}$ together with the additivity of the integral and the fact that the coefficients $C_{k}(\Delta,u,V)$ are uniquely determined. ∎