stylistic:style_transfer, linguistic:paraphrasing
Example: prime factorization

Another proof by complete induction uses the hypothesis that the statement holds for all smaller n {\displaystyle n} more thoroughly. Consider the statement that "every natural number greater than 1 is a product of (one or more) prime numbers ", which is the " existence " part of the fundamental theorem of arithmetic. For proving the induction step, the induction hypothesis is that for a given n > 1 {\displaystyle n>1} the statement holds for all smaller n > 1 {\displaystyle n>1}. If m {\displaystyle m} is prime then it is certainly a product of primes, and if not, then by definition it is a product: m = n 1 n 2 {\displaystyle m=n_{1}n_{2}}, where neither of the factors is equal to 1; hence neither is equal to m {\displaystyle m}, and so both are greater than 1 and smaller than m {\displaystyle m}. The induction hypothesis now applies to n 1 {\displaystyle n_{1}} and n 2 {\displaystyle n_{2}}, so each one is a product of primes. Thus m {\displaystyle m} is a product of products of primes, and hence by extension a product of primes itself.