Expression for Integer as Product of Primes is Unique/Proof 2

Theorem
Let $n$ be an integer such that $n > 1$.

Then the expression for $n$ as the product of one or more primes is unique up to the order in which they appear.

Proof
Suppose $n$ has two prime factorizations:
 * $n = p_1 p_2 \dots p_r = q_1 q_2 \dots q_s$

where $r \le s$ and each $p_i$ and $q_j$ is prime with $p_1 \le p_2 \le \dots \le p_r$ and $q_1 \le q_2 \le \dots \le q_s$.

Since $p_1 \mathop \backslash q_1 q_2 \dots q_s$, it follows from Euclid's Lemma for Prime Divisors that $p_1 = q_j$ for some $1 \le j \le s$.

Thus:
 * $p_1 \ge q_1$

Similarly, since $q_1 \mathop \backslash p_1 p_2 \dots p_r$, from Euclid's Lemma for Prime Divisors:
 * $q_1 \ge p_1$

Thus, $p_1 = q_1$, so we may cancel these common factors, which gives:
 * $p_2 p_3 \cdots p_r = q_2 q_3 \dots q_s$

This process is repeated to show that:
 * $p_2 = q_2, p_3 = q_3, \ldots, p_r = q_r$

If $r < s$, we arrive at $1 = q_{r+1} q_{r+2} \cdots q_s$ after canceling all common factors.

But by [Divisors of One]], the only divisors $1$ are $1$ and $-1$.

Hence $q_{r+1}, q_{r+2}, \ldots, q_s$ cannot be prime numbers

From that contradiction it follows that $r = s$.

Thus:
 * $p_1 = q_1, p_2 = q_2, \ldots, p_r = q_s$

which means the two factorizations are identical.

Therefore, the prime factorization of $n$ is unique.