Consider the following premises. For all x_10, if x_10 listens to a song, then x_10 is writing a letter. For all x_10, if x_10 writes a letter, then x_10 is a janitor. For all x_10, if x_10 takes a plane, then x_10 is a janitor. For all x_10, x_10 is bored. There is at least one x_10 for which x_10 will take a plane or x_10 will listen to a song. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_10 for which x_10 is a janitor and x_10 is bored.
 From the fact that for all x_10, if x_10 listens to a song, then x_10 is writing a letter, and that for all x_10, if x_10 writes a letter, then x_10 is a janitor we can infer that for all x_10, if x_10 listens to a song, then x_10 is a janitor via universal transitivity. From the fact that for all x_10, if x_10 takes a plane, then x_10 is a janitor, that for all x_10, if x_10 listens to a song, then x_10 is a janitor, and that there is at least one x_10 for which x_10 will take a plane or x_10 will listen to a song we can infer that there is at least one x_10 for which x_10 is a janitor via existential disjunction elimination. Finally, from the fact that there is at least one x_10 for which x_10 is a janitor, and that for all x_10, x_10 is bored we can infer that there is at least one x_10 for which x_10 is a janitor and x_10 is bored via existential conjunction. Therefore, the answer is yes.