Theorem 7.5: (Distributivity of Multiplication of Natural Numbers Across Subtraction) Let a, b, and c be natural numbers with c < b. Then

a (b - c ) = ab - ac

Proof: Since c < b, b - c is a natural number by Theorem 7.3. Let

d = b - c

Then

b = d + c

by Theorem 7.3, so

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

by Theorem 6.10, the Distributivity of Multiplication of Natural Numbers across Addition. The result then follows from Theorem 7.3.