Substitution of indifferent lotteries
= Substitution of indifferent lotteries
{title2=$A\sim B\ \Longrightarrow\ pA+(1-p)C\sim pB+(1-p)C$}
Apply the <independence axiom for lottery preferences> in both weak directions of the <indifference relation>. At $p=0$ the two mixtures are identical. Repeating this substitution and using <transitivity> lets one replace any finite collection of indifferent component <lotteries>, after which <reduction of compound lotteries> combines their outcome <probabilities>.