Theorem 6.10: ( The Distributive Property for Natural Numbers): Let a, b, and c be natural numbers. Then

a (b + c ) = ab + ac

Proof: By Theorem 6.2, we can find two disjoint sets, B and C, where #(B ) = b and #(C ) = c. Then a x B and a x C will also be disjoint since

by Theorem 2.9, the distributivity of Cartesian products across intersections.

= a x { }

since B and C are disjoint

= { }

by Theorem 5.8. Then by Theorem 6.1

and

But

by Theorem 2.8, the Distributivity of Cartesian products across unions, so by Theorem 5.4, the well definedness of cardinality,

a (b + c ) = ab + ac