Consider the following premises. There is at least one z for which if z writes a letter, then z is playing tennis. For all z, if z writes a letter, then z is playing tennis. For all z, if z plays tennis, then z is writing a letter. For all z, z will not write a letter. Can we infer the following from them? There is at least one z for which z will not play tennis.
 From the fact that for all z, if z plays tennis, then z is writing a letter, and that for all z, if z writes a letter, then z is playing tennis we can infer that for all z, z will play tennis if and only if z will write a letter. From the fact that for all z, z will play tennis if and only if z will write a letter we can infer that for all z, if z plays tennis, then z is writing a letter. From the fact that there is at least one z for which if z writes a letter, then z is playing tennis, and that for all z, if z plays tennis, then z is writing a letter we can infer that there is at least one z for which z will write a letter if and only if z will play tennis. Finally, from the fact that there is at least one z for which z will write a letter if and only if z will play tennis, and that for all z, z will not write a letter we can infer that there is at least one z for which z will not play tennis. Therefore, the answer is yes.