Theorem 2.5: (Distributivity of Intersections across Unions) Let A, B, and C be three sets. then

.

Proof: Let

Then, first,

by Definition 1.3 and Definition 1.4. But in either case,

Then either

so

Conversely suppose that

This says that either

In either case

and we also have that

This says that

Since we have containment both ways we conclude that

by Definition 1.2.