Integers form Unique Factorization Domain

Theorem
The integers $\left({\Z, +, \times}\right)$ form a unique factorization domain.

Proof
Follows directly from:


 * $(1) \quad$ the fundamental theorem of arithmetic
 * $(2) \quad$ the fact that $\left({\Z, +, \times}\right)$ is an integral domain
 * $(3) \quad$ the definitions complete factorization and equivalent factorizations.