Consider the following premises. For all x_6, if x_6 takes a plane, then x_6 is cooking. There is at least one x_6 for which if x_6 were a politician, then x_6 is cooking. For all x_6, x_6 will write a letter or x_6 is a politician. For all x_6, if x_6 were a cashier, then x_6 is taking a plane. For all x_6, x_6 will not write a letter or x_6 is a cashier. Can we infer the following from them? There is at least one x_6 for which x_6 will cook.
 From the fact that for all x_6, x_6 will write a letter or x_6 is a politician, and that for all x_6, x_6 will not write a letter or x_6 is a cashier we can infer that for all x_6, x_6 is a politician or x_6 is a cashier. From the fact that for all x_6, if x_6 were a cashier, then x_6 is taking a plane, and that for all x_6, if x_6 takes a plane, then x_6 is cooking we can infer that for all x_6, if x_6 were a cashier, then x_6 is cooking. Finally, from the fact that there is at least one x_6 for which if x_6 were a politician, then x_6 is cooking, that for all x_6, if x_6 were a cashier, then x_6 is cooking, and that for all x_6, x_6 is a politician or x_6 is a cashier we can infer that there is at least one x_6 for which x_6 will cook. Therefore, the answer is yes.