Consider the following premises. It is not cloudy. It is raining. It is not raining or Mary is cooking. Can we infer the following from them? Mary is cooking.
 From the fact that it is raining we can infer that it is raining or it is cloudy. From the fact that it is raining or it is cloudy, and that it is not raining or Mary is cooking we can infer that it is cloudy or Mary is cooking. Finally, from the fact that it is cloudy or Mary is cooking, and that it is not cloudy we can infer that Mary is cooking. Therefore, the answer is yes.