Proposition 4

*Let $a, b_1, \ldots, b_n$ be ideals of a ring $A$. If $a$ is relatively prime to each of the $b_i$ ($1 \leq i \leq n$), it is relatively prime to $b_1 b_2 \ldots b_n$.

Let $p$ be a prime ideal of $A$. If $p$ contains $a$ and $b_1 b_2 \ldots b_n$ it contains one of the $b_i$ (no. 1, Proposition 1), which is absurd since $a$ and $b_i$ are relatively prime.