We keep the notation in the proof of Proposition 2.21. In particular, $s^{\otimes e}=\lambda\sigma$ with $\lambda$ in the fraction field of ${\mathcal{A}}$, and $\mathscr{H}(p)=H$. We verify that

$$\frac{\log\|s(p)\|}{\log|\varpi|}=\frac{\log|\lambda(p)|}{e\log|\varpi|}=\frac{\log|\operatorname{N}_{H/K}(\lambda(p))|}{e[H:K]\log|\varpi|}=\frac{{\operatorname{ord}}_{\varpi}(\operatorname{N}_{H/K}(\lambda(p)))}{e[H:K]}$$

and

$$({\widetilde{p}}\cdot\operatorname{div}(s^{\otimes e}))=\deg(\rho_{\ast}(\operatorname{div}({\widetilde{p}}^{\ast}s^{\otimes e})))=\deg(\rho_{\ast}(\operatorname{div}(\lambda(p))))\\ =\deg(\operatorname{div}(\operatorname{N}_{H/K}(\lambda(p))))={\operatorname{ord}}_{\varpi}(\operatorname{N}_{H/K}(\lambda(p))),$$

which proves the statement. ∎