Corollary 1

Suppose that $A_i \not\subset A_j$ for $i \neq j$. For every family of elements $a_i \in A_i$ ($1 \leq i \leq n$), there exists $x \in B$ such that $x \equiv a_i \pmod{m(A_i)}$ for $1 \leq i \leq n$.

Since the $p_i$ are maximal ideals of $B$, $A_i/m(A_i) = B_{p_i}/p_iB_{p_i} = B/p_i$ and it may therefore be assumed that $a_i \in B$ for all $i$. The corollary then follows from the fact that the canonical mapping from $B$ to $\prod_{i=1}^n (B/p_i)$ is surjective (Chapter 11, § 1, no. 2, Proposition 5).