Appendix A1, PDF p. 28 (printed p. 22) — the profit threshold exceeds the price threshold: e_π > e_p

Let $t>0$ and $0<\mu<t$. Then

1. $t\sqrt{t^2-\mu^2}>(t-\mu)^2$ holds if and only if $\mu(t-\mu)(4t^2-3t\mu+\mu^2)>0$;
2. the profit threshold exceeds the price threshold: $e_p<e_\pi$, where $e_p=2(t-\mu)$.

Item 2 is the clause "where $e_\pi>e_p$" of Proposition 2: there is a range $(e_p,e_\pi)$ of exclusive features in which network effects lower the price but still raise the profit.

Formalization Note Under the stated hypotheses both sides of item 1 are true; item 1 records the paper's equivalence step. The network effect is strict, $\mu>0$: at $\mu=0$, $e_\pi=e_p=2t$.