Proposition 2

With the hypotheses of Proposition 1 suppose further that $\mathbf{A}_i \not\subset \mathbf{A}_j$ for $i \neq j$. Then the $p_i$ are distinct maximal ideals of $B$ and every maximal ideal of $B$ is equal to one of the $p_i$.

If $p_i \subset p_j$ for $i \neq j$, $\mathbf{A}_i = B_{p_i} \supset B_{p_j} = \mathbf{A}_j$. It is then sufficient to apply Chapter II, § 3, no. 5, Corollary to Proposition 17.