Complete induction proof that every $n > 1$ can be written as a product of primes
For every natural $1<n<N$, if $n$ is not prime we can write it as a product of primes.
Let $N$ be composite, i.e. $N=a\cdot b$, with $1<a, b<N$.
Both $a$ and $b$ can be written either as a single prime or as a product of primes.
In any case, $N$ can be written as a product of primes, so that:
For every natural $1<n<N\color{red}{+1}$, if $n$ is not prime we can write it as a product of primes.