# 证明该环同态是前面提到的环同构

The restriction $A=\mathcal O_X(X)\to \mathcal O_X(X_{f_i})$ maps $f_i$ to a unit. So it induces a homomorphism $\psi:A_{f_i}\to \mathcal O_X(X_{f_i})$ which is an isomorphism.

$\varphi|_{X_{f_i}}:X_{f_i}\to D(f_i)$
$\because X_{f_i}\to D(f_i)\to \mathrm{Spec}\,A$的复合与$X_{f_i}\to X\to \mathrm{Spec}\,A$的复合是相同的，
$\therefore (\varphi|_{X_{f_i}})^\sharp (D(f_i)):A_{f_i}\to \mathcal O_X(X_{f_i})$与$\psi:A_{f_i}\to \mathcal O_X(X_{f_i})$是相同的。