I certanly could not understand what the problem is.
Could you please possibly explain what is the problem statement?
Hmm. Start out as follows:
Given.
There is a vector such that for any vector , we have .
Since is a vector, there is another vector, call it , such that . [EDIT Here]
We add this vector to both sides: .
Since addition is associative and commutative, we may rewrite as follows:
. Can you finish?
The usual method to prove something is unique is to assume there is another solution, and prove it is equal to the one you already found. So, assume there is another solution, , and show that . I think you'll need the uniqueness of the zero vector, and the uniqueness of inverses to show the result.