The following definitions will be used in the proof of the statement. an integer n is even if, and only if, n equals twice some integer. an integer n is odd if, and only if, n equals twice some integer plus 1. symbolically, for any integer, n n is even ⇔ n = 2k for some integer k n is odd ⇔ n = 2k + 1 for some integer k prove the following statement. the difference of any two odd integers is even. proof: let m and n be any odd integers. by definition of odd, there are integers r and s so that m can be expressed in terms of r and n can be expressed in terms of s as follows: m = n = write m − n in terms of r and s and factor out a 2 to obtain m − n = . now is an integer because of integers are integers. therefore, m − n = 2 · (an integer), and so m − n is by definition of .