Consider the following premises. For all x_11, if x_11 writes a letter, then x_11 is not working. For all x_11, if x_11 climbs a mountain, then x_11 is working. There is at least one x_11 for which x_11 will climb a mountain. For all x_11, x_11 will write a letter or x_11 will play a game. Can we infer the following from them? There is at least one x_11 for which x_11 will climb a mountain.
That is one of the premises. Therefore, the answer is yes.