How do you distribute AND and OR?

Hi all,

I'm working through a book on set theory and I haven't gotten very far yet. I'm sort of struck by the looseness of the proofs for the associative, distributive, and commutative laws provided by the book and I think this stems from the fact that I'm not sure how to relate / distribute the ANDs and ORs of written logic.

Example:

If x is an element of (A intersect B) union C, then x is an element of A AND x is an element of B, OR x is an element of C.

Why does the proof jump from this kind of formulation to: (A U C) intersect (B U C)?

Why is the OR back distributed over the AND?

Any help clarifying my confusion would be much appreciated.

Thanks,

Ultros