Theorem 2.9: (The Distributivity of Cartesian Products Across Intersections). Let A, B, and C be three sets. If

and

Proof: (Right Distributivity) Let

Then by Definition 1.9,

Since

then

Since also

then

Thus,

Conclude that

by Definition 1.1. Conversely, let

Since

by Definition 1.9.

Since also,

then

Thus,

so

Conclude that

Since we have containment both ways,

by Definition 1.2.

(Left Distributivity) Let

Then

then

and

Thus,

Conclude that

Conversely, let

Since

Since also,

then

Thus,

so

Conclude that

Since we have containment both ways,

by Definition 1.2