I'm having trouble proving the distributive law for [r] [s] [t] in Zn (the integers mod n)
i.e. [r]([s] + [t]) = [r][s] + [r][t]
I did this so far.
(r+m1*n)[(s+m2*n)+(t+m3*n)] = (r+m1*n)(s+m2*n)+(r+m3*n)(t+m4*n)
Here is an example how you prove it is well-defined.
Say you wanted to prove $\displaystyle [a]+[b]=[a+b]$.
Say $\displaystyle [a]=[c]$ and $\displaystyle [b]=[d]$.
This means $\displaystyle a\equiv c ~ (n)$ and $\displaystyle b\equiv d ~ (n)$.
Thus, $\displaystyle a+b\equiv c+d ~ (n)$ so $\displaystyle [a+c]=[b+d]$.
This shows that $\displaystyle +$ is well-defined.
Modify this argument to the problem you have.