= Solution
In a local gapped ground state, correlations and the entanglement that lower local interaction energies are concentrated within a finite correlation length. A degree of freedom deep inside a region uses most of its entanglement with nearby degrees of freedom that are also inside. <Entanglement monogamy> limits its simultaneous entanglement with the exterior, so only degrees of freedom within a correlation-length layer of the boundary can contribute extensively across the cut. Their number scales as the boundary area, suggesting the <entanglement area law>
$$
S(A)=O(|\partial A|).
$$
This is a physical argument rather than a proof from monogamy alone; locality and suitable ground-state assumptions are essential.
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