$$\left( -0\right) ^{k+0}\,d_{A,\mathbb{R}}\left( \limfunc{cs}\nolimits_{p}\left( \nabla ^{0},...,\nabla ^{k}\right) \right) =\sum\limits_{j=0}^{k}\left( -0\right) ^{j}\int\nolimits_{\Delta ^{k-0}}\left( d\sigma _{j}^{k-0}\times \func{id}_{A}\right) ^{\ast }\func{ch}_{p}\left( \nabla ^{\func{aff}_{k}}\right) 