I attempted to prove this fact using the well-ordering principle, here is my initial work:
Every natural number greater than 1 is either prime or can be expressed as a unique product of primes.
Proof Let $n\in\mathbb{N}$ be greater than 1, and composite.
Existence: Define
$T=\{a\in\mathbb{N}: \exists b\in \mathbb{N}(ab=n) \land 1<ab<n\}$
Notice that $T$ is non-empty, since $n$ is composite. Then, by the well-ordering principle, $T$ has a smallest element: call it $\alpha$. If $\alpha$ is prime, we are done, so we assume $\alpha$ is composite.
Since $\alpha$ is composite, it can be written in terms of two integers $j,k$ such that $1<j,k<\alpha$. But then $\alpha b=jkb=j(kb)$ where $j<\alpha$. But this contradicts the fact that $\alpha$ is the smallest element of $T$. Thus $\alpha$ must be prime.
But I am stuck at this point. I proved that every natural number greater than 1 has a prime factor, but not that all its factors are prime. How can I proceed to show that its other factors must be prime as well?
