Theorem 7.7: If a and b are natural numbers then a + b > a with equality if and only if b = 0.

Proof: By Definition 6.1 a + b is the b th successor of a. If b = 0, then a + b is the 0th successor of a which is simply a. Note that the successor of a,

and

by Theorem 1.2, so

a < a + 1

by Definition 4.5. By Theorem 4.5c, the transitivity of <,

a < a + b

However, note that

but

so if b > 0 then