Editing Mathematical induction (section)

==== Example: prime factorization ====
Another proof by complete induction uses the hypothesis that the statement holds for ''all'' smaller <math>n</math> more thoroughly. Consider the statement that "every [[natural number]] greater than 1 is a product of (one or more) [[prime number]]s", which is the "[[Fundamental theorem of arithmetic#Existence|existence]]" part of the [[fundamental theorem of arithmetic]]. For proving the induction step, the induction hypothesis is that for a given <math>m>1</math> the statement holds for all smaller <math>n>1</math>. If <math>m</math> is prime then it is certainly a product of primes, and if not, then by definition it is a product: <math>m = n_1 n_2</math>, where neither of the factors is equal to 1; hence neither is equal to <math>m</math>, and so both are greater than 1 and smaller than <math>m</math>. The induction hypothesis now applies to <math>n_1</math> and <math>n_2</math>, so each one is a product of primes. Thus <math>m</math> is a product of products of primes, and hence by extension a product of primes itself.