Theorem 6.3: (The Commutative Property for Addition of Natural Numbers) Let a and b be natural numbers. Then

Proof: By Theorem 6.2 we can find two disjoint sets S, and T such that

and

Then by Theorem 6.1

and

However, by Theorem 2.1, the commutativity of unions of sets,

so by Theorem 5.4, the well definedness of cardinality,