Theorem 7.8: If a and b are non zero natural numbers, then

with equality if and only if b = 1.

Proof: If b = 1, then by Theorem 6.9 The Identity Principle for Multiplication,

but if b > 1 then its predecessor, b - 1 > 1 and

by Theorem 4.10b

by Theorem 6.10

by Theorem 7.7.