Consider the following premises. forall x: Q_1(x) -> S(x). forall x: S(x) -> R(x). exists x: U(x) or ~S(x). forall x: R(x) -> Q_1(x). Can we infer exists x: R(x) -> U(x) from them?
 From exists x: U(x) or ~S(x) we can infer exists x: S(x) -> U(x). From forall x: R(x) -> Q_1(x), forall x: Q_1(x) -> S(x) we can infer forall x: R(x) -> S(x). From forall x: R(x) -> S(x), forall x: S(x) -> R(x) we can infer forall x: R(x) <-> S(x). From forall x: R(x) <-> S(x) we can infer forall x: R(x) -> S(x). Finally, from forall x: R(x) -> S(x), exists x: S(x) -> U(x) we can infer exists x: R(x) -> U(x). Therefore, the answer is yes.