4. Analyze the proof of the theorem and mention one rule of inference used in the proof.

Theorem Let n ? ?. If 7n + 6 is odd then n is odd.

Proof

Let us assume that n is not odd, that is, n is even. Then we have that n = 2k for some k ? ?. So

7n + 6 = 7(2k) + 6

= 2(7k) + 6

= 2(7k + 3)

From here we conclude that 7n + 6 is even in contradiction with the premise of the theorem. Therefore, if 7n + 6 is odd then n must be odd.