= Solution
We prove the forward direction of the <finite-generator criterion for a natural uniform algebra>. Suppose that $A$ is a <natural uniform algebra>, and let $f_1,\ldots,f_n$ have no common zero on $K$. The <ideal>
$$
J=f_1A+\cdots+f_nA
$$
is all of $A$. Otherwise it is contained in a <maximal ideal> $M$, whose associated <character of an algebra> vanishes on all the $f_j$. Since every <algebra character> is an <evaluation character> under naturality, this would give a common zero, a contradiction. Therefore $1\in J$, meaning that
$$
\boxed{\sum_{j=1}^nf_jg_j=1\quad\hbox{for suitable }g_j\in A.}
$$
Here are the Banach-algebra details behind the <algebra character> associated with $M$. A proper <ideal> cannot have dense closure: if it contained an element within distance less than one of $1$, the <Neumann series> would make that element invertible, forcing the <ideal> to contain $1$. Thus the closure of a proper <ideal> is proper. A <maximal ideal>, obtained from <Zorn's lemma>, is therefore closed. Its quotient is a complex unital <Banach algebra> in which every nonzero element is invertible. The <Gelfand-Mazur theorem> identifies this quotient with $\mathbb C$, so the quotient map is the required <algebra character>. No closedness of the originally generated <ideal> $J$ is needed.
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