Direct sum, modules question

So I have attached the picture.

I have two questions.

1) In the green box, is that some rule? Or can you just do that?

2) I don't understand anything after the red line. can anyone shed some light? Why do they take x to be an element in that? Is it a misprint?

1) becomes clear if you write out the definition of each module:

$\frac{A}{A \cap B}+\frac{B}{A \cap B} = \{a+(A\cap B) + b +(A\cap B)|a \in A, b \in B\}=\{a+b+(A\cap B)|a \in A, b \in B\}=\frac{A+B}{A \cap B}$

2) is just a misprint, it should say $x \in \frac{A}{A \cap B} \cap \frac{B}{A \cap B}$

Sorry to bother again. But Im stuck on the very last bit >_>

The bit which says (b-a) e AnB ==> a=b-(b-a) e AnB

I cannot see why this implies that a is an element of AnB. I mean, if b were an element of AnB, then I could conclude it by additive closure, but I dont know that b is in AnB, and if I did, I wouldnt need a anyways >_>

Thanks

The notes you've posted are quite brief at that point - basically, they're showing that a is in A and a is in B, so therefore a is in AnB.

You've got a in A by definition, and since you know b is in B, and you've deduced b-a is in B, you get that a is in B by writing a = b-(b-a).