 A further application of the axioms lets us establish that these preferences over lotteries satisfy three important properties: (i) a “Mixture Weak Order” condition, requiring the preferences over lotteries to be transitive and complete, (ii) a “Mixture Continuity” condition, the details of which are not of importance here and finally (iii) an “Independence” condition, which, alongside the ordering condition, will be the focus of considerable discussion in what follows.