Consider the following premises. forall x_13: Q_13(x_13) -> Q_2(x_13). exists x_13: P_2(x_13). forall x_13: ~Q_2(x_13) or S(x_13). forall x_13: P_2(x_13) -> Q_2(x_13). Can we infer exists x_13: R_4(x_13) or S(x_13) from them? If possible, name the inference rules being used at each step.
 From exists x_13: P_2(x_13) we can infer exists x_13: P_2(x_13) or Q_13(x_13) via existential addition. From forall x_13: P_2(x_13) -> Q_2(x_13), forall x_13: Q_13(x_13) -> Q_2(x_13), exists x_13: P_2(x_13) or Q_13(x_13) we can infer exists x_13: Q_2(x_13) via existential disjunction elimination. From exists x_13: Q_2(x_13) we can infer exists x_13: Q_2(x_13) or R_4(x_13) via existential addition. Finally, from exists x_13: Q_2(x_13) or R_4(x_13), forall x_13: ~Q_2(x_13) or S(x_13) we can infer exists x_13: R_4(x_13) or S(x_13) via existential resolution. Therefore, the answer is yes.