In the Appendix to Ivan Niven's book "Numbers: Rational and Irrational", he proves the Fundamental Theorem of Arithmetic (FToA) without using Euclid's Lemma that if a prime divides a product, then it divides one of the factors of the product. Niven's proof instead uses well-ordering.
He assumes that m is the smallest positive integer with two different prime factorizations, say
$$ m=p_1 p_2 p_3 \ldots p_r \qquad \text{and} \qquad m=q_1 q_2 q_3 \ldots q_s $$
Typically, Euclid's Lemma would now be invoked repeatedly to say that each prime in one factorization of a number must also occur in any other, giving a contradiction. (Or more typically, this idea is rewritten as a direct proof.)
Niven instead proceeds saying that the two factorizations cannot have a prime in common, since if they did, we could assume without loss of generality that p_1 = q_1. Then $m/p_1$ would be a positive integer smaller than $m$ with the two different factorizations
$$ m/p_1 = p_2 p_3 \ldots p_r \qquad \text{and} \qquad m/p_1 = q_2 q_3 \ldots q_s, $$
a contradiction.
So without losing generality, we may assume $p_1 < q_1$. Niven proceeds to show that the number $(q_1 - p_1) q_2 q_3 \cdots q_s$ is a positive integer smaller than $m$ with two different prime factorizations --- one containing $p_1$ as a factor and the other not. This contradiction proves the uniqueness part of the FToA.
My question is, who originated this proof?