Rural hospitals theorem, Part 2

Consider a many-to-one matching market with a finite set $D$ of doctors and a finite set $H$ of hospitals, where hospital $h$ has $q_h$ positions and all preferences are strict. For a matching $\mu$, write $\mu(h)\subseteq D$ for the set of doctors assigned to $h$. Let $\mu$ and $\nu$ be stable matchings and let $h\in H$ be a hospital with an empty position under $\mu$, i.e. $|\mu(h)|<q_h$. Then

This is Part 2 of the rural hospitals theorem (Roth 1986): an under-subscribed hospital receives exactly the same set of doctors in every stable matching.

Formalization Note Uses the model RuralHospitals.Market (definition RuralHospitals_Market): finite types D (doctors) and H (hospitals); strict preferences given by injective rank functions on Option H / Option D (smaller rank = preferred, none = unmatched / empty position); responsive hospital preferences; capacities capacity : H → ℕ. An assignment is a map μ : D → Option H, and assigned μ h is the finite set of doctors placed at h.