Corollary 1

*For every family of elements $\gamma_i \in \Gamma_i$ ($1 \leq i \leq n$), there exists $x \in K$ such that $v_i(x) = \gamma_i$ ($1 \leq i \leq n$).

We may assume that $A_i \neq K$ for all $i$. Then, there exists for all $i$ an $a_i \in K$ such that $v_i(a_i) = \gamma_i$ and an $a_r \in \Gamma_r$ such that $\gamma_i < a_r$. We apply Theorem 1 to these elements $a_i$: there exists $x \in K$ such that $v_i(x - a_i) > v_i(a_i)$; whence, as $x = a_i + (x - a_i)$, $v_i(x) = v_i(a_i) = \gamma_i$ ($§ 3$, no. 1, Proposition 1).