Proof: Let $\alpha\in A_{c}^{n-1,n}(N_{\mathbb{R}})$. By Stokes’ formula in Proposition 3.5, we have

$$\delta_{({\mathscr{C}},m)}(d^{\prime}\alpha)=\int_{\partial({\mathscr{C}},m)}\alpha=\sum_{\sigma}m_{\sigma}\sum_{\rho\subset\sigma}\int_{\rho}\langle\alpha;\omega_{\rho,\sigma}\rangle_{\{n\}}=\sum_{\rho}\int_{\rho}\langle\alpha;\sum_{\sigma\supset\rho}m_{\sigma}\omega_{\rho,\sigma}\rangle_{\{n\}},$$

where $\rho$ (resp. $\sigma$) ranges over all elements of ${\mathscr{C}}$ of dimension $n-1$ (resp. $n$). Suppose now that $\sum_{\sigma\supset\rho}m_{\sigma}\omega_{\rho,\sigma}\in N_{\rho}$ for some $n-1$-dimensional $\rho\in{\mathscr{C}}$. Recall that we may view $\alpha$ as a multilinear map $N_{{\mathbb{R}}}^{2n-1}\rightarrow C^{\infty}(N_{\mathbb{R}})$ which is alternating in the first $n-1$ arguments and also alternating in the last $n$ arguments. But an alternating $n$-linear map on a vector space of dimension $n-1$ is zero and hence the restriction of $\langle\alpha;\sum_{\sigma\supset\rho}m_{\sigma}\omega_{\rho,\sigma}\rangle_{\{n\}}$ to $\rho$ is zero. Then the above display proves (a) $\Rightarrow$ (b).