Consider the following premises. forall x_10: Q_5(x_10) -> S(x_10). forall x_10: S(x_10) or ~Q_10(x_10). forall x_10: Q_10(x_10) or P_5(x_10). forall x_10: P_5(x_10) -> Q_5(x_10). Can we infer forall x_10: S(x_10) or T(x_10) from them? If possible, name the inference rules being used at each step.
 From forall x_10: S(x_10) or ~Q_10(x_10) we can infer forall x_10: Q_10(x_10) -> S(x_10) via universal rewriting. From forall x_10: P_5(x_10) -> Q_5(x_10), forall x_10: Q_5(x_10) -> S(x_10) we can infer forall x_10: P_5(x_10) -> S(x_10) via universal transitivity. From forall x_10: Q_10(x_10) -> S(x_10), forall x_10: P_5(x_10) -> S(x_10), forall x_10: Q_10(x_10) or P_5(x_10) we can infer forall x_10: S(x_10) via universal disjunction elimination. Finally, from forall x_10: S(x_10) we can infer forall x_10: S(x_10) or T(x_10) via universal addition. Therefore, the answer is yes.