Addition of positive and negative of a object in vector space = 0 related question

Sorry if I am posting too frequently. I am learning linear algebra all by myself without

any teacher or TA or without going to college. It's very fun but sometimes the book

gets overly complicated in here and there. That's why these postings.

Now back to the problem. Suppose in a vector space:

And addition and subtraction are defined as:

And a axiom is given:

For each in , there is an object in , called a negative of , such that

We have to prove if the above vector space satisfies this axiom.

So I did it like this:

Now:

Now how do I get ? The book says that it satisfies this axiom.

But don't know how to reach it.

Can anyone kindly help me how to get to this relationship?