Consider the following premises. There is at least one x_11 for which if x_11 were a politician, then x_11 is playing a game. For all x_11, x_11 is a janitor or x_11 is a politician. For all x_11, x_11 will not play a game or x_11 will cook. For all x_11, if x_11 were a janitor, then x_11 is playing a game. Can we infer the following from them? There is at least one x_11 for which x_11 is a soldier or x_11 will cook.
 From the fact that for all x_11, if x_11 were a janitor, then x_11 is playing a game, that there is at least one x_11 for which if x_11 were a politician, then x_11 is playing a game, and that for all x_11, x_11 is a janitor or x_11 is a politician we can infer that there is at least one x_11 for which x_11 will play a game. From the fact that there is at least one x_11 for which x_11 will play a game we can infer that there is at least one x_11 for which x_11 will play a game or x_11 is a soldier. Finally, from the fact that there is at least one x_11 for which x_11 will play a game or x_11 is a soldier, and that for all x_11, x_11 will not play a game or x_11 will cook we can infer that there is at least one x_11 for which x_11 is a soldier or x_11 will cook. Therefore, the answer is yes.