I am reading this explanation of zkSnarks written by Maksym Petkus - http://www.petkus.info/papers/WhyAndHowZkSnarkWorks.pdf
In section 4.5, the pdf explains how to represent the following operations
$a$ x $b = r1$
$r1$ x $c = r2$
as $l(x)r(x) - o(x)$ where $l(x)$ is the left operand polynomial, $r(x)$ is the right operand polynomial & $o(x)$ is the output polynomial.
Here if you see that $a$ is used only once in the LHS of the first set of operations - it's used only in the first one (i.e. $a$ x $b = r1$) - it doesn't feature in the 2nd one.
In 4.6, he moves on to how to do the same thing when $a$ repeats. i.e.
$a$ x $b = r1$
$a$ x $c = r2$
Here $a$ is present in both operations
So he says
Nevertheless, because our protocol allows prover to set any coefficients to a polynomial, he is not restricted from setting different values of $a$ for different operations (i.e., represented by some x)
This freedom breaks consistency and allows prover to prove the execution of some other program which is not what verifier is interested in. Therefore we must ensure that any variable can only have a single value across every operation it is used in
Further ahead in 4.6.1 he says
Consequently, if a verifier needs to enforce the prover to set the same value in all operations, then it should only be possible to modify the proportion and not the individual coefficients.
I am unable to understand this - the coefficients of $l(x)$ come from all the operations (you find $l(x)$ by using something like Lagrange's Polynomial Interpolation on both operations). So how can the prover use different coefficients for different operations. Can some explain with an example?