Regular functions on a normal variety are cut out in codimension one

Given: AC, the algebraically closed field $k$, the irreducible normal classical variety $X$ with function field $k(X)$, a rational function $\phi \in k(X)$, and a finite affine chart $U\subseteq X$ with coordinate ring $A=k[U]$.

$A$ is a Noetherian integrally closed domain; a rational function is regular at a point $y$ exactly when it lies in the local ring ${\mathcal{O}}_{X,y}$, and on the affine chart this local ring is the localisation $A_{{\mathfrak{m}}_y}$ (Normality is checked on affine open charts, Normal points and normal varieties, A rational function is regular at a point exactly when it lies in the local ring, The local ring at a point of an affine variety is the localization at its maximal ideal). AC is used here.

Every Noetherian integrally closed domain satisfies Serre's condition $(S_2)$, and a Noetherian domain with $(S_2)$ equals the intersection of its height-one localisations inside its fraction field (normal domain implies s two, r one s two intersection of height one localisations); the intersection is read inside $k(X)=\mathrm{Frac}(A)$ (The function field of an irreducible classical affine variety).

All affine charts share the field $k(X)$ (Integral classical varieties in the compatible affine-atlas register, Compatible affine charts of an integral classical variety have one function field). A prime $\mathfrak{p}$ on a chart defines the prime-local ring $A_{\mathfrak{p}}$ along its irreducible subvariety. We use height one to express codimension one, rather than adjoining nonclosed points to the classical point set.