§2.1, p. 8 — three equilibria between threshold prices, two at a threshold

In a market with In/Out inverse demand, fix $p_{\min}\le p\le p_{\max}$. Each of the three branches of $P$ has exactly one intersection with the price $p$, denoted $q_{\mathrm{out}}$, $q_{\mathrm{mid}}$, and $q_{\mathrm{in}}$. These intersections exhaust the equilibrium quantities:

For $p_{\min}<p<p_{\max}$ they satisfy $q_{\mathrm{out}}<q_{\mathrm{mid}}<q_{\mathrm{in}}$. At $p=p_{\min}$ the out and middle intersections coincide and the in intersection is larger; at $p=p_{\max}$ the middle and in intersections coincide and the out intersection is smaller. Thus the boundary prices have two distinct equilibrium quantities.

This result supplies the branch quantities used by Proposition 1 and rules out other feasible fixed points.

Formalization Note Branch intersections are predicates on closed intervals, so the repeated equilibrium at a threshold belongs to two branches. Continuity of $P$ is not assumed separately; it must follow from the market assumptions.