Proof: Using definitions from Section 7.2 one sees immediately that the statement of the Lemma follows from the equality $p_{\Omega^{can}}(1+\eta)=1$, which is straightoforward: $p_{\Omega^{can}}(1+\eta)=exp\left(Res(\Omega^{can}\,\log(1+\eta))\right)=1\in{\cal O}_{K}^{\times}$. $\,\blacksquare$