In this article, rigorous mathematical analysis is used to prove that the necessary condition for multiple steady-state solutions to exist in two-stage separation process problems involving ternary mixtures is the tendency of the mixture to exhibit a second liquid phase. The unrealistic solution multiplicity that exists in these problems when phase splitting is not included in the model equations used to solve them, vanishes when phase splitting is included. The standard specification of reflux ratio and bottoms flow rate has been investigated.