Use the fact that $\displaystyle \gcd(a,n)=1$ if and only if there exist integers $\displaystyle r,s$ with $\displaystyle ra+sn=1.$ Thus

$\displaystyle ax\equiv1\,(\bmod\,n)$ has solution in $\displaystyle x$

$\displaystyle \iff\ ax+kn\,=\,1$ for some integer $\displaystyle k$

$\displaystyle \iff\ \gcd(a,n)\,=\,1$

What you want to prove is that if $\displaystyle r\equiv i\,(\bmod\,n)$ and $\displaystyle r'\equiv j\,(\bmod\,n)$ then $\displaystyle r+r'\equiv i+j\,(\bmod\,n).$ Your proof is essentially correct, but it might be better to write the $\displaystyle r$ and $\displaystyle r'$ on the left-hand side instead.

(NB: There is no need to assume $\displaystyle r\le q$ or $\displaystyle r'\le q'.)$