Consider the following premises. For all z_2, if z_2 writes a letter, then z_2 is climbing a mountain. For all z_2, if z_2 climbs a mountain, then z_2 is writing a letter. There is at least one z_2 for which if z_2 writes a letter, then z_2 is climbing a mountain. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z_2 for which if z_2 writes a letter, then z_2 is climbing a mountain.
That is one of the premises. Therefore, the answer is yes.