Theorem 7.10: (The Cancellation Property for Multiplication of Natural Numbers) If a, b, and c are natural numbers with a not 0 and ab = ac, then b = c.

Proof: Assume that a is not 0 and ab = ac. Then by Theorem 7.4

Use the Distributive Property of Subtraction, Theorem 7.5, to factor out the a

Since a is not 0, it follows from the integrity of multiplication of natural numbers, Theorem 7.6, that

Now by Theorem 7.3 t7-3.html,