Commentary on Proposition IX.14¶

In other words, a number can be resolved into prime factors in only one way.

Let $a$ be the least number measured by each of the prime numbers $b, c, d, \dots k$.

If possible, suppose that $a$ has a prime factor $p$ different from $b, c, d, \dots k$.

Let

$$a = p \cdot m.$$

Now $b, c, d, \dots k$, measuring $a$, must measure one of the two factors $p, m$. [VII. 30]

They do not, by hypothesis, measure $p$ ; therefore they must measure $m$, a number less than $a$ : which is contrary to the hypothesis.

Therefore $a$ has no prime factors except $b, c, d, \dots k$.