Here is the axiom
Simplification
For each $j = 1,...,J$, let $L_j$ be the following simple lottery $$L_j = [p_1^j(A_1),...,p_K^j(A_k)],$$ and let $\hat{L_j}$ be the following compound lottery $$\hat{L_j} = [q_1(L_1),...,q_J(L_J)].$$ For each $k = 1,2,...,K$ define $$r_k = q_1p_k^1 + ... + q_Jp_k^J;$$ this is the overall probability that the outcome of the compound lottery $\hat{L}$ will be $A_k$. Consider the simple lottery $$L = [r_1(A_1),...,r_k(A_k)].$$ Then $$\hat{L} \approx_i L$$
The equivalence relation $L_a \approx_i L_b$ means that a player $i$ is indifferent between $L_a$ and $L_b$.
I am having trouble understanding where the expression for $r_k$ comes from. Could someone help me?
$p_j^k$ is the probability that the outcome of $L_j$ will be $A_k$. $q_j$ is the probability that the outcome of the compound lottery $\hat{L}$ is $L_j$. So $$r_k = \sum_i(\text{probability that outcome of compound lottery is } L_i)×(\text{outcome of lottery } L_i \text{ is } A_k) $$