Rothschild-Stiglitz (RS) Separating Equilibrium
I recommend reading 'Informational Equilibrium' by John Riley (1979). It discusses this setting more generally. How the "insurance model" of Rotschild and Stiglitz is a special case of this is presented on pg. 335, and then Theorem 3 on pg. 343 is what you need. The proof is not trivial, but - in my opinion - well presented.