Please prove that the preference relation satisfies the independence axiom (the yellow highlight).

Suppose that a preference relation 2 on the space of simple lotteries _ can be represented by a utility function U(.) that has the expected utility form, that is, a vNM expected utility function U(L) = uip, + ... + UNPN for every simple lottery L = (P1 . . .. . PN) EL. Show that this preference relation 2 satisfies the independence axiom. A preference relation _ over lotteries satisfies the independence axiom if, for any three lotteries L, L', L" E L, and for a E (0,1), we have LZ L' if and only if al + (1 - a)L" Z aL' + (1 -a)L". VNM - Von Neumann Morgenstern