Why do we need the Well-Ordering Property in the proof of Fundamental Theorem of Arithmetic?

Question

The standard proof of the fundamental theorem of arithmetic uses the well-ordering property of natural numbers. The contradiction argument claims that "By the Well-Ordering Principle, there is a smallest integer with this property" (meaning can be written as products of primes in more that one way).

My question is, why do we need to assume this? Why cannot we just build the contradiction claim as "Let's assume there is an integer which can be written as products of primes in more than one way" and continue to show the contradiction just following the exact same steps?

Answer 1

Without the well ordering property each number with multiple factorizations could be the product of some prime and another number with multiple factorizations.

Answer 2

A contradiction could be, that if there exist a least element, it produces a lower least element in a given set. In any set with a least element, this either has to stop eventually, or the solutions can no longer be labelled because some aren't even in the set, and none can be labelled the least, because we can't put them in bijection with ordinals. ex.

$$i^2=-1\\(-1)\cdot 1=-1\\(-i)^2=-1$$ Without a sense of ordering, we can't label any one of these the least factorization of -1 . Similarly, we can't call something the smallest counterexample, if it produces a smaller counterexample, where smallest is replacing first in order of a list of counterexamples ordered by place in underlying set.

The whole point of prime factorization is to an extent, to find a labeling of a number, written using the least factorizable numbers as bases, without an ordering, how do we tell which numbers those are ?