Corollary 2

Suppose that $A_i \not\subset A_j$ for $i \neq j$. There exist elements $x_i$ ($1 \leq i \leq n$) of $K$ such that $v_i(x_i) = 0$ and $v_j(x_i) > 0$ for $i \neq j$.

For each index $i$ apply Corollary 1 to the family $(a_i)$ such that $a_i = 1$ and $a_j = 0$ for $j \neq i$.