Theorem 7.9: If a, b, and c are nonzero natural numbers with b < c, then

ab < ac.

Proof: If b < c, then d = c - b is a nonzero natural number, by Definition 7.1, and c = b + d by Theorem 7.3. Then ad is a nonzero natural number.

ac = a (b + d ) = ab + ad

by Theorem 6.10, and so

ab < ac

by Theorem 7.7.