Factorization of Natural Numbers within 4 n + 1 not Unique

Theorem
Let:
 * $S = \set {4 n + 1: n \in \N} = \set {1, 5, 9, 13, 17, \ldots}$

be the set of natural numbers of the form $4 n + 1$.

Then not all elements of $S$ have a complete factorization by other elements of $S$ which is unique.

Proof
Proof by Counterexample:

Consider the number:
 * $m = 693 = 3^2 \times 7 \times 11$

Thus:
 * $m = 9 \times 77 = 21 \times 33$

We have that:

The divisors of these numbers are as follows:

Thus $693$ has two different complete factorizations into elements of $S$.

Hence the result.