Corollary 2

Let $h : A \to A'$ be a ring homomorphism such that $A'$ is integral over $A$, $a$ and $p$ two ideals of $A$ such that $a \subset p$ and $a'$ an ideal of $A'$ lying above $a$. Suppose that $p$ is prime. Then there exists a prime ideal $p'$ of $A'$ lying above $p$ and containing $a'$.

If $h_1 : A/a \to A'/a'$ is the homomorphism derived from $h$ by taking quotients, $h_1$ is injective by hypothesis and $A'/a'$ is integral over $A/a$ ($§ 1$, no. 1, Proposition 2); then there exists a prime ideal $p'/a'$ of $A'/a'$ (p' prime in $A$) lying above $p/a$ (Theorem 1) and $p'$ is the required ideal.