We consider the rational function $\varpi^{-a+i}t$. Since the support of $\operatorname{div}(\varpi^{-a+i}t)$ does not contain the component $E_{i}$ nor any of the components $F_{i,j}$, we have that

$$|\varpi^{-a+i}t(\xi_{i})|=|\varpi^{-a+i}t(\xi_{i,j})|=1.$$

Since $t=\chi^{1}$, we deduce, using equation (5.4), that

$${\operatorname{val}}_{K}(\xi_{i})=\frac{-\log|\chi^{1}(x_{i})|}{\lambda_{K}}=\frac{-\log|\varpi^{a-i}|}{-\log|\varpi|}=a-i.$$

∎