Construction of prior in Bayesian Statistics

I was trying to read this book on Bayesian Statistic and have I have trouble understanding the part in orange in the bottom image:

I think I get what is defined in equation (3.9) and how having $$R = X-AX$$ could give us $$X-AX \sim \pi_{mod.error}(r)$$ but I don't get how we can have the prior distribution of $$x$$ from that, maybe knowing what is "hidden" behind proportional term could help. Source : (Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing, Springer, 2007)

It is actually a type of issue I encountered several times in this type of examples ( like in interpolation noise-free data ) and I never know how to deal with those cases when we have an $$Ax=R$$ and we want to express the prior of $$x$$ given that the distribution of $$R$$ and the value of $$A$$ are known. I think the example above is quite representative. Any advice on how I could understand that?

$$X$$ follows the following distribution by definition : $$X \sim \pi_{prior}(x)$$. Also by definition, $$R \sim \pi_{mod.error}(r)$$.
We also defined $$X = AX+R$$ so $$R = X-AX$$ implying that $$\pi_{mod.error}(r) = \pi_{mod.error}(x-Ax)$$.
maybe rewriting $$R = X-AX$$ as $$R = X (I -A)$$ allows you to see that $$R$$ and $$X$$ are proportionnal up to the term $$I - A$$.
so we find effectively that $$\pi_{mod.error}(x-Ax)$$ and $$\pi_{prior}(x)$$ are proportionnal up to the term $$I - A$$.