Independence and Reduction Axioms
You can find in many textbooks (for example Mas-Collel, Winston and Green) that the Independence of irrelevant alternatives axiom implies that preferences over lotteries are linear with respect to the probabilities of each event. 

Then, a compound lottery is of the form: $\mathcal{L}''=\alpha\mathcal{L}+(1-\alpha)\mathcal{L}'$, If $A$ and $B$ are the outcomes of the simple lottery $\mathcal{L}$ and $C$ and $D$ are the outcomes of the other simple lottery, $\mathcal{L}'$. Then, the compound lottery can be seen as a simple lottery with four outcomes, given the linearity in probabilities: $\mathcal{L}''=\alpha\left(\beta A+(1-\beta)B\right)+(1-\alpha)\left(\gamma C+(1-\gamma)D\right)$.