I am struggling to understand the proof of the second step in the Expected Utility Theorem, particularly the part that deals with preferences over weighted sums of lotteries. The statement I am trying to prove is:
For the best possible lottery $ L_h $ and the worst possible lottery $ L_l $, the compound lottery $ bL_h + (1-b)L_l $ is preferred over $ aL_h + (1-a)L_l $ if and only if $ b > a $.
I've attempted to prove it by contradiction but have encountered difficulties in rigorously showing how having 'more' of $ L_h $ in one lottery over another implies it should be preferred. I of course understand the intuition, but I am struggling with the formal proof using the independence axiom.
Thank you in advance for your help