Compatible affine charts of an integral classical variety have one function field

Let U,V be two nonempty affine opens of X. Their overlap is nonempty by F2. F3 supplies a nonempty principal open $W\subseteq U\cap V$ in U. By F4 it is affine, and F1 identifies its locally regular structure with that inherited from V. Thus W is also an affine open of V. F5 identifies both k(U) and k(V) with k(W), defining a comparison $c_{UV}:k(U)\to k(V)$.

For another choice $W^{\mathrm{\prime }}$, the intersection $W\cap W^{\mathrm{\prime }}$ is nonempty open in U. Choose a nonempty principal open T of U in that intersection. It is affine with the inherited structure in W and $W^{\mathrm{\prime }}$. F5 says all field comparisons are restriction maps, and F6 says two fraction values that agree after restriction are equal. Both candidate comparisons therefore agree after passing to k(T), an isomorphic field, so they are equal.

For three nonempty affine opens U,V,Z, choose a nonempty principal open T of U inside $U\cap V\cap Z$, possible by applying F2 twice and then F3. By step 2.1 it may be used in every pairwise comparison. All three identifications then pass through k(T), so $c_{VZ}\circ c_{UV}=c_{UZ}$, $c_{UU}$ is the identity, and $c_{VU}=c_{UV}^{-1}$. Restrictions commute by the same construction. An isomorphism of varieties carries common affine opens and their regular-function restrictions to common affine opens; its pullbacks commute pointwise with restrictions and hence with their field extensions. This proves compatibility with isomorphisms and defines a single field independent of the chosen chart.