Corollary

Let $p$ (resp. $p'$) be a prime ideal of $A$ (resp. $A'$) of height 1 and $v$ (resp. $u'$) the essential valuation of $A$ (resp. $A'$) corresponding to it. For $p'$ to lie above $p$, it is necessary and sufficient that the restriction of $v'$ to $K$ be equivalent to $v$.

The valuation $v'$ is equivalent to the extension of an essential valuation $w$ of $A$ (Proposition 12). Let $q = p' \cap A$, which is a prime ideal of $A$ of height 1. For the restriction of $v'$ to $K$ to be equivalent to $v$, it is necessary and sufficient that $w = v$ and hence that $q = p$.