(a) Prove the equivalence (5.25), including the inequalities, for real numbers \(A, B, C\) that fulfill \(A

(b) Prove \((1-\alpha) \mathbb{E}(u, P)+\alpha \mathbb{E}(u, R)=\mathbb{E}(u,(1-\alpha) P+\alpha R)\) for two lotteries \(P\) and \(R\), which is the second equation in (5.30).