Affineness from a finite principal cover

Write $1={\sum }_i{g}_i{f}_i$. At every point some $f_i$ is a unit, so the $U_i$ cover $X$. Set $B_i=\mathrm{\Gamma }(U_i,{\mathcal{O}}_X)$. The intersection $U_i\cap U_j$ is the principal open defined by $f_j{\mathrm{\mid }}_{U_i}$ in $U_i$, hence affine, and F2 gives its ring by localization.

F4 identifies $R$ as the kernel of the difference of restriction maps ${\prod }_i{B}_i\to {\prod }_{i,j}\mathrm{\Gamma }(U_i\cap U_j,{\mathcal{O}}_X)$, a homomorphism of $R$-modules. Fix $k$ and localize this kernel at $f_k$. Localization preserves kernels by F3: apply exactness to the kernel-image short exact sequence and to the inclusion of the image in the target. It commutes with these finite products, because a common denominator exists for every finite tuple.

By F2 each localized factor is the ring of sections on its intersection with $U_k$. By F4 their kernel is $\mathrm{\Gamma }(U_k,{\mathcal{O}}_X)$, since the opens $U_i\cap U_k$ cover $U_k$. Thus restriction induces $R_{f_k}\cong B_k$ as rings; multiplicativity follows from restriction and fraction multiplication, not merely module exactness.

F1 supplies the canonical map $j:X\to \mathrm{Spec}R$ induced by the identity of $R$. Its inverse image of $D(f_k)$ is $X_{f_k}$, and on this open it is the isomorphism from step 3.1. The $D(f_k)$ cover $\mathrm{Spec}R$ because the $f_k$ generate 1; local inverse maps agree and glue to the inverse of $j$. If $n=0$ then $R=0$, which forces $X=\mathrm{\varnothing }$ since each point has a nonzero local ring. For $n=1$, $f_1$ is a unit and $U_1=X$. Zero sections merely contribute empty charts.