Consider the following premises. There is at least one z_3 for which z_3 is not a lawyer or z_3 will write a letter. For all z_3, z_3 is not a cashier or z_3 is a bartender. For all z_3, z_3 is a lawyer or z_3 is a cashier. For all z_3, z_3 will not write a letter. Can we infer the following from them? For all z_3, z_3 is not a bartender.
 From the fact that for all z_3, z_3 is a lawyer or z_3 is a cashier, and that there is at least one z_3 for which z_3 is not a lawyer or z_3 will write a letter we can infer that there is at least one z_3 for which z_3 is a cashier or z_3 will write a letter. From the fact that there is at least one z_3 for which z_3 is a cashier or z_3 will write a letter, and that for all z_3, z_3 is not a cashier or z_3 is a bartender we can infer that there is at least one z_3 for which z_3 will write a letter or z_3 is a bartender. Finally, from the fact that there is at least one z_3 for which z_3 will write a letter or z_3 is a bartender, and that for all z_3, z_3 will not write a letter we can infer that there is at least one z_3 for which z_3 is a bartender, which contradicts that For all z_3, z_3 is not a bartender. Therefore, the answer is no.