I am trying to study the proof of the Chinese remainder theorem from the book Dummit and Foote (Section 7.6, page 266, third edition). The theorem itself is to show that $R/(A_1A_2\ldots A_k) \cong R/A_1 \times \ldots \times R/A_k) $ if the ideals $A_1, \ldots,A_k$ are comaximal, i.e $A_i +A_j=R$ if $i \neq j$. The proof is by induction and I understand the case $k=2$. Now they are trying to show that $A_1$ and $(A_2 A_3 \ldots A_k)$ are comaximal and I don't fully understand this argument. It is claimed that for $i \in \{ 2,3,\ldots,k\}$ there are $x_i \in A_1$ and $y_i \in A_i$ such that $x_i + y_i =1$ (I guess this follows from the hypothesis that the ideals are comaximal, and that $x_i +y_i \equiv y_i (\mbox{mod} A_1)$ (which is because $x_i + y_i + A_1 = y_i + A_1$ since $x_i \in A_1$ ). Now they say $1= (x_2+y_2) \ldots (x_k+y_k)$ is an element in $A_1 + (A_2 \ldots A_k)$.
I would be grateful if somebody can kindly elaborate on this last statement. I also don't see where they use the fact that $x_i +y_i \equiv y_i (\mbox{mod} A_1)$
Thank you.