= Solution
The <localization of global sections on a principal open> gives $A_{f_i}=\Gamma(X_{f_i},\mathcal O_X)$. Each of these rings is a <finitely generated algebra> because $X_{f_i}$ is an <affine variety>. Choose finite generators of every $A_{f_i}$ and write them as $a_{ij}/f_i^{e_{ij}}$ with $a_{ij}\in A$. Let $B$ be the $k$-subalgebra of $A$ generated by all $f_i,g_i,a_{ij}$. It is a <finitely generated algebra>, and $B_{f_i}=A_{f_i}$ since these localizations contain the chosen generators and $f_i^{-1}$.
For any integer $N\geq1$, raise $\sum_i g_if_i=1$ to the power $r(N-1)+1$. Every resulting monomial contains some $f_i^N$, so it gives a <unit-ideal identity for powers>
$$
1=\sum_i c_if_i^N,\qquad c_i\in B.
$$
For an arbitrary $a\in A$, equality $A_{f_i}=B_{f_i}$ implies $f_i^{N_i}a\in B$ for some $N_i$: multiply by an additional power if equality of localized fractions requires it. Choose a common $N$ and the identity above. Then $a=\sum_i c_i(f_i^Na)\in B$. Consequently $A=B$, so $A$ is a <finitely generated algebra>. It is a <reduced ring>, since a <nilpotent element> global <regular function> has zero germ everywhere on the reduced <variety> $X$. This is <finite generation from finitely many principal localizations>.
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