Proth Numbers

Let n be a positive natural number. Let us note that n− 1 is natural. Let n be a non trivial natural number. Observe that n− 1 is positive. Let x be an integer number and n be a natural number. Let us observe that xn is integer. Let us observe that 1n reduces to 1. Let n be an even natural number. Let us observe that (−1) reduces to 1. Let n be an odd natural number. One can verify that (−1) reduces to −1. Now we state the propositions: (1) Let us consider a positive natural number a and natural numbers n, m. If n ­ m, then an ­ am. (2) Let us consider a non trivial natural number a and natural numbers n, m. If n > m, then an > am. The theorem is a consequence of (1).