Consider the following premises. P_1(a_1). Can we infer forall z_4: ~(P_1(z_4) or P_3(z_4)) from them? If possible, name the inference rules being used at each step.
 From P_1(a_1) we can infer exists z_4: P_1(z_4) via existential generalization. Finally, from exists z_4: P_1(z_4) we can infer exists z_4: P_1(z_4) or P_3(z_4) via existential addition, which contradicts forall z_4: ~(P_1(z_4) or P_3(z_4)). Therefore, the answer is no.