the inequality $$\min \{ c_\pi -b_\pi'+1 , (c_\pi +d_\pi)- (a_\pi+b_\pi) \} \leq \min \{ c_\pi - b_\pi', c_\pi - b_\pi\}.$$ Since $b_\pi' < b_\pi$ and $d_\pi \leq a_\pi$, the left-hand side is $(c_\pi - b_\pi) + (d_\pi - a_\pi)$ and the right-hand side is $c_\pi -b_\pi$, and the inequality follows.

The integer $x$ given in the previous result satisfies a simpler set of congruences, which we describe in the following lemma.

\[system\] Assume ${\mathrm{GCD}}(p,n) = {\mathrm{GCD}}(p',n)$ and ${\mathrm{GCD}}(q,n) = {\mathrm{GCD}}(q',n)$ and let $x$ be the integer given in Lemma \[=&gt;&lt;-lem\]. Then both of the following hold:

1.  $ \tfrac{rn}{pq} x + \tfrac{r}{q} $ and $  \tfrac{r}{p} x $ are both divisible by $\frac{r}{q'}$.

2.  For all primes $\pi$ dividing both $\frac{rn}{pq}$ and $\frac{r}{q}$, we have $  \tfrac{rn}{pq\pi} x + \tfrac{r}{q\pi} \not \equiv 0 \modu \tfrac{r}{q'})$.

To prove that (1) holds, it suffices to show $\tfrac{rn}{pq} x + \tfrac{r}{q} \equiv   \tfrac{r}{p} x \equiv 0 \modu \pi^{c_\pi - b_\pi'})$ for all primes $\pi$ dividing $rn$. Deriving this set of congruences from Lemma \[=&gt;&lt;-lem\] is straightforward.

Let $\pi$ be a prime dividing both $\frac{rn}{pq}$ and $\frac{r}{q}$. To complete the lemma, it is enough to show that $\tfrac{rn}{pq\pi} x + \tfrac{r}{q\pi} \not \equiv 0 