Is this a sufficient proof for why -(-a) equals a?

So, on the final day of my math class, my professor gave us a few simple challenges to think about, and one of them was to try to prove why -(-a) equals a. The key here was prove, not explain, so I can't just say turn yourself around twice.

It got me thinking for a few days as how to do this because it seems so simple so I don't even know where to start. This is what I eventually settled on.

The - symbol in math seems to have 2 main uses. If used between 2 numbers, like a-b, it can be thought of as a + -b. But that brings into question: what does -b mean?

What I eventually settled on was we can think of -b as the number x that satifies the equation b + x = 0. This seems to be a defintion I can't find issues with, but I'm not entirely sure.

Now, what does -(-a) mean? By the previous defintion, -(-a) would be the number x that satifies the equation x + -a = 0. Now, since -a is intended to be the number we add to a to get 0, it should also follow that if we add a to -a, we get 0. Thus, it should follow x is a.

Therefore, -(-a) equals a.

I'm not exactly sure if this counts as a proof, but I had fun thinking about how to prove it so even if this turns out to have holes, I'm happy. If real analysis is like this, I can't wait to take it in like 4 or so years.