Bass' paper on Gorenstein rings

Question

I am currently reading the paper On the ubiquity of Gorenstein rings by Hyman Bass.
I found difficulty to understand the proof of Proposition (7.2).
Under the the following setting:
$A$: commutative Noetherian ring with Krull $\dim A\leq1$, no nilpotent elements
$A'$: integral closure of $A$ is a finitely generated $A$-module
$K$: $S^{-1}A$ where $S=\{$non zero divisor in $A\}$
$M$: f.g. $A$-module

Suppose $M\neq 0$ is reflexive $A$-module. Then either $M$ has a nonzero projective direct summand or else $M$ is an $A_1$-module for some $A\subsetneq A_1 \subset A'$.

The proof is quite long. Part that I don't understand is when the writer tried to use Chinese Remainder Theorem.
Could somebody point out for me on what object shall I apply Chinese Reminder Theorem?