= Solution
The supplied row-hook formula is
$$
H_i(\lambda)=\{1,\ldots,h_i\}
\setminus\{h_i-h_j:i<j\leq m\}.
$$
Consequently $h\in H_i(\lambda)$ exactly when $h_i-h\geq0$ and $h_i-h$ is not one of $h_{i+1},\ldots,h_m$. Since $h_i-h\leq h_i<h_j$ for $j<i$, this is equivalent to $h_i-h\notin X$. We have proved the <hook criterion in a beta set>
$$
\boxed{h\in H_i(\lambda)\Longleftrightarrow h_i-h\geq0\text{ and }h_i-h\notin X}.
$$
If $ef$ is a hook length, the beta-set interpretation gives a bead at some position $b$ and a gap at $b-ef$. In the finite progression
$$
b,b-e,b-2e,\ldots,b-fe,
$$
the first position is occupied and the last is empty. Some consecutive pair is therefore a bead followed by a gap. Their distance is $e$, so the criterion gives a hook of length $e$. This proves the <divisor closure of hook lengths>.
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