Consider the following premises. exists y: S_2(y) -> T(y). forall y: T(y) -> S(y). forall y: S_1(y). forall y: S_2(y). Can we infer exists y: S_1(y) from them?
 From exists y: S_2(y) -> T(y), forall y: T(y) -> S(y) we can infer exists y: S_2(y) -> S(y). From exists y: S_2(y) -> S(y), forall y: S_2(y) we can infer exists y: S(y). From forall y: S_1(y), exists y: S(y) we can infer exists y: S_1(y) and S(y). Finally, from exists y: S_1(y) and S(y) we can infer exists y: S_1(y). Therefore, the answer is yes.