Theorem 7.1

Theorem 7.1: (The Cancellation Property of Addition of Natural Numbers) If a, b, and c are natural numbers and if

a + c = b + c

then

a = b.

Proof: By Theorem 4.6, we can assume that a < b. Define a function

f (x ) = x + c

Since x + c is the c th successor of x, f is actually a composition of c successor functions, the first from b to b + 1, the next from b + 1 to b + 2 etc. Theorem 5.6 tells us that a successor function is one to one, and By Theorem 3.2, the composition of one to one functions is also a one to one function, so by Definition 3.5, a = b.