Hi, I hope this is the correct forum, if not my apologies

I've been confused with this for a while and have decided I am really stuck.

I have to prove that |-a| = |a| using the absolute value definition which was given to us as:

Now I understand the following by definition:

if a>0 then

if a<0 then

But with I get really confused! My lecturer explained it as follows:

if , then,

and if , then

Firstly, Is this because originally we treat a as positive, but because it is made negative within the absolute value, then we treat it as if it is negative, thus -(-a)?

Then with a < 0, is it so because, we originally treat it as a<0, thus -(-a) but then as a negative, thus -a? Also with the RHS, -(-a) then -a?

Some clarification on this would be really appreciated, here is my attempt at proving it:

if , then

if , then

As you can see, I'm missing some small fact to make this true or I am making a really silly mistake somewhere as this seems like it should be very easy!