Consider the following premises. For all x_5, if x_5 plays a game, then x_5 is writing a letter. For all x_5, x_5 will play a game or x_5 will make tea. For all x_5, if x_5 writes a letter, then x_5 is playing a game. For all x_5, if x_5 makes tea, then x_5 is writing a letter. Can we infer the following from them? There is at least one x_5 for which x_5 will not write a letter.
 From the fact that for all x_5, if x_5 plays a game, then x_5 is writing a letter, and that for all x_5, if x_5 writes a letter, then x_5 is playing a game we can infer that for all x_5, x_5 will play a game if and only if x_5 will write a letter. From the fact that for all x_5, x_5 will play a game if and only if x_5 will write a letter we can infer that for all x_5, if x_5 plays a game, then x_5 is writing a letter. Finally, from the fact that for all x_5, if x_5 plays a game, then x_5 is writing a letter, that for all x_5, if x_5 makes tea, then x_5 is writing a letter, and that for all x_5, x_5 will play a game or x_5 will make tea we can infer that for all x_5, x_5 will write a letter, which contradicts that There is at least one x_5 for which x_5 will not write a letter. Therefore, the answer is no.