Agreement of the domain and prime-quotient valuative formulas

Fix a valuation overring $V$ of a domain $R$, with fraction field $K=\mathrm{Frac}(R)$. Its fraction field is $K$, since $R\subseteq V\subseteq K$. Any overring $W$ of $V$ in $K$ is again a valuation ring by [F1]. By [F3], it has maximal ideal ${\mathfrak{m}}_W$ consisting of its nonunits; this is prime because a product is a unit exactly when both factors are units. Hence $\mathfrak{p}={\mathfrak{m}}_W\cap V$ is prime. Every $s\in V\setminus \mathfrak{p}$ is a unit in $W$, so $V_{\mathfrak{p}}\subseteq W$. Conversely, if $w\in W\setminus V$, then $w^{-1}\in V$ by [F1], and it is a unit of $W$, so $w^{-1}\mathrm{\notin }\mathfrak{p}$ and $w\in V_{\mathfrak{p}}$. Thus $W=V_{\mathfrak{p}}$. Each $V_{\mathfrak{p}}$ is a valuation ring, and [F3] recovers $\mathfrak{p}$ as the contraction of its unique maximal ideal. These correspondences reverse inclusions: inclusion of primes reverses the denominator sets, and inclusion of overrings preserves units and therefore reverses the contractions of their nonunits. They preserve strictness by being inverse bijections. The endpoints are $V=V_{{\mathfrak{m}}_V}$ and $K=V_{(0)}$. Finite prime chains in $V$ can be extended by these endpoints, so their length supremum equals that of finite strict chains of valuation overrings from $V$ to $K$. Taking the supremum over all $V$ containing $R$ yields $\delta (R)=c(R)$.

Put $\rho (D)={\sup }_{\mathfrak{p}\in \mathrm{Spec}D}\delta (D\mathrm{/}\mathfrak{p})$. These are domain inputs by [F2]. Applying step 1.1 to each quotient gives $\rho (D)=c_{\ast }(D)$, and applying it to $D$ gives $\delta (D)=c(D)$. For every finite integer $n\ge 0$, [F5] identifies both $c(D)\le n$ and $c_{\ast }(D)\le n$ with exactly the same test $\dim D[T_1,\dots ,T_k]\le n+k$ for all $k\ge 0$. The hypotheses hold because $D$ is nonzero and [F6] supplies the imported classical setting. Therefore $\delta (D)$ and $\rho (D)$ have exactly the same finite upper bounds. Both are in $\{0,1,\dots ,\mathrm{\infty }\}$: fraction fields provide nonempty valuation-overring families, and $(0)$ is a prime of $D$. Equal finite upper bounds determine the same value, including infinity, proving the equality.