I need to prove that $\displaystyle A-B = A - (A\cap B)$

Using a Venn diagram, this proof is so trivial but how do I prove this mathematically?

So far I have

$\displaystyle A-B = A\cap B^{C}$

$\displaystyle =A\cap(S-B)$ where S is the sample space

But now I'm stuck...