Proposition 4

Let $v_1, \ldots, v_n$ ($n \geq 2$) be pairwise dependent valuations on the same field $K$. Then the rings $A_1, \ldots, A_n$ of $v_1, \ldots, v_n$ generate a subring $\delta \ K$ distinct from $K$.

For $n = 2$ Proposition 4 follows from Definition 1. Suppose it holds for $n - 1$ valuations. Then there exists a subring A of K distinct from K and containing $A_1, \ldots, A_{n-1}$; there also exists a subring B $\neq K$ containing $A_{n-1}$ and $A_n$. As A and B contain $A_{n-1}$, they are comparable with respect to inclusion (§ 4, no. 1, Corollary to Proposition 1). The greater of these two therefore contains all the $A_i$.