Corollary

Let A and B be two valuation rings of K, where $A \subset B$; let $A' = h_B(A)$, which is a valuation ring of $\kappa(B)$. The residue field $\kappa(A')$ of $A'$ is canonically isomorphic to the residue field $\kappa(A)$ of A and the place $h_A$ associated with A is the composition $h_{A'} \circ h_B$ of the places associated with A and B.

Since the local ring $A'$ is a quotient of the local ring $A$, their residue fields are canonically isomorphic and the equation $h_A(x) = h_{A'}(h_B(x))$ holds for $x \in A$. On the other hand, if $x \in B - A$, then $h_B(x) \notin A'$ and the two sides of the equation are equal to $\infty$; the same is true if $x \in K - B$.