Proof . Part 1) follows from the fact that $\rho_{\delta}$ is a closed current. Part 2) follows from the fact that $R_{\delta}$ changes the supports of $Z_{i},i=1,2$ by $O(\delta)$. To prove part 3) one observes that the operators $id$ and $(R_{\delta})_{|\Omega^{\ast}(Y)}$ preserve the space of smooth forms $\Omega^{\ast}(Y)$, and $\rho_{\delta}$ is cohomologous to $[diag]$. $\blacksquare$