Regular functions on a nonempty open embed in the affine function field

For $s\in {\mathcal{O}}_X(U)$, choose a nonempty quotient neighbourhood $W\subseteq U$ with $s=a\mathrm{/}b$ and $b$ nowhere zero on $W$. Then $b\mathrm{\ne }0$ in $A$, so $a\mathrm{/}b\in k(X)$. If $s=c\mathrm{/}d$ on another nonempty quotient neighbourhood $W^{\mathrm{\prime }}$, F1 says $W\cap W^{\mathrm{\prime }}$ is nonempty dense. There $ad-bc=0$ pointwise. Its zero locus is closed, so it vanishes on X, and F2 gives $ad=bc$ in A. F5 therefore identifies the two fractions.

Define the image of s to be that uniquely determined fraction. Near a point in a common nonempty quotient neighbourhood for s and t, F4 gives the sum and product by cross multiplication; those are exactly the fraction-field operations. Constants map to themselves, so this is a $k$-algebra map. If s maps to 0, every local quotient $a\mathrm{/}b$ has $a=0$ by F5 and step 1.1, hence s vanishes on each quotient neighbourhood and therefore on all U. Thus the map is injective.

If $V\subseteq U$ is nonempty open, a quotient neighbourhood for $s{\mathrm{\mid }}_V$ is also a nonempty open subdomain of U. Step 1.1 shows that it computes the same fraction as s. This proves compatibility with restrictions and with every permitted nonempty quotient presentation.