Consider the following premises. It is cloudy if and only if Sarah is working. If Thomas writes a letter, then it is cloudy. If it is cloudy, then Thomas is writing a letter. Can we infer the following from them? The fact that Sarah works does not imply that Thomas is writing a letter.
 From the fact that it is cloudy if and only if Sarah is working we can infer that if Sarah works, then it is cloudy. From the fact that if it is cloudy, then Thomas is writing a letter, and that if Thomas writes a letter, then it is cloudy we can infer that it is cloudy if and only if Thomas is writing a letter. From the fact that it is cloudy if and only if Thomas is writing a letter we can infer that if it is cloudy, then Thomas is writing a letter. Finally, from the fact that if Sarah works, then it is cloudy, and that if it is cloudy, then Thomas is writing a letter we can infer that if Sarah works, then Thomas is writing a letter, which contradicts that The fact that Sarah works does not imply that Thomas is writing a letter. Therefore, the answer is no.