Corollary 3

Let K be a field, K' an extension of K and A a valuation ring of K. The integral closure of A in K' is the intersection of the valuation rings V' of K such that V' $\cap K = A$.

Theorem 1 (c) show that, if V' is a valuation ring of K', V' $\cap K$ is a valuation ring of K and V' dominates V' $\cap K$. For V' to dominate A, it is necessary and sufficient that V' $\cap K$ dominate A and therefore be equal to it.