Consider the following premises. For all x_2, x_2 will play a game or x_2 will sleep. For all x_2, if x_2 plays a game, then x_2 is fast. There is at least one x_2 for which x_2 is not fast. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_2 for which x_2 will sleep.
 From the fact that for all x_2, if x_2 plays a game, then x_2 is fast, and that there is at least one x_2 for which x_2 is not fast we can infer that there is at least one x_2 for which x_2 will not play a game via existential modus tollens. Finally, from the fact that for all x_2, x_2 will play a game or x_2 will sleep, and that there is at least one x_2 for which x_2 will not play a game we can infer that there is at least one x_2 for which x_2 will sleep via existential disjunctive syllogism. Therefore, the answer is yes.