Hey Linseykathleen.

I don't know about how your professor/lecturer/whatever expects of proofs but if you need a really rigorous one try considering that if A is a subset of B then A intersection B = A for all A and B such that A is a subset of B.

This way if (A and C) is a subset of (B and D), then (A and C) and (B and D) = (A and C).