Consider the following premises. For all x_6, x_6 will cook. For all x_6, x_6 will write a letter. There is at least one x_6 for which if x_6 goes running, then x_6 is not writing a letter. For all x_6, if x_6 cooks, then x_6 is working. Can we infer the following from them? There is at least one x_6 for which x_6 will not go running.
 From the fact that for all x_6, if x_6 cooks, then x_6 is working, and that for all x_6, x_6 will cook we can infer that for all x_6, x_6 will work. From the fact that for all x_6, x_6 will write a letter, and that for all x_6, x_6 will work we can infer that for all x_6, x_6 will write a letter and x_6 will work. From the fact that for all x_6, x_6 will write a letter and x_6 will work we can infer that for all x_6, x_6 will write a letter. Finally, from the fact that there is at least one x_6 for which if x_6 goes running, then x_6 is not writing a letter, and that for all x_6, x_6 will write a letter we can infer that there is at least one x_6 for which x_6 will not go running. Therefore, the answer is yes.