Going up theorem

Statement
This result is sometimes called going up and sometimes lying over and going up. It is a stronger version of lying over.

Suppose $$f:R \to S$$ is an injective homomorphism of commutative unital rings, such that $$S$$ is an integral extension of $$R$$. Suppose $$P$$ is a prime ideal of $$R$$, and $$Q_1$$ is an ideal of $$S$$ such that $$f^{-1}(Q_1) \subset P$$. Then, there exists a prime ideal $$Q$$ containing $$Q_1$$, such that $$f^{-1}(Q) = P$$.

Proof
This follows from lying over, applied to the injective map $$R/f^{-1}(Q_1) \to S/Q_1$$.