if , then has a unique sol.
By Bezout, there exist such that
Now, we can write that as:
I don't know about that. What we are trying to prove is that there is a unique value for (up to congruence modulo ) satisfying (where are fixed). I don't think that what you have done gives us the uniqueness.
Try this: suppose there exist such that . We try to prove that .
It's fairly simple: what happens if you multiply the congruence by ?