Consider the following premises. For all x_8, if x_8 cooks, then x_8 is writing a letter. Michael is cooking. For all x_8, if x_8 writes a letter, then x_8 is cooking. Can we infer the following from them? Michael is writing a letter.
 From the fact that for all x_8, if x_8 cooks, then x_8 is writing a letter, and that for all x_8, if x_8 writes a letter, then x_8 is cooking we can infer that for all x_8, x_8 will cook if and only if x_8 will write a letter. From the fact that for all x_8, x_8 will cook if and only if x_8 will write a letter we can infer that Michael is cooking if and only if Michael is writing a letter. From the fact that Michael is cooking if and only if Michael is writing a letter we can infer that if Michael cooks, then Michael is writing a letter. Finally, from the fact that if Michael cooks, then Michael is writing a letter, and that Michael is cooking we can infer that Michael is writing a letter. Therefore, the answer is yes.