Existence of Greatest Common Divisor/Proof 3

Proof
From Integers form Integral Domain, we have that $\Z$ is an integral domain.

From Elements of Euclidean Domain have Greatest Common Divisor, $a$ and $b$ have a greatest common divisor $c$.

This proves existence.

From Ring of Integers is Principal Ideal Domain, we have that $\Z$ is a principal ideal domain.

Suppose $c$ and $c'$ are both greatest common divisors of $a$ and $b$.

From Greatest Common Divisors in Principal Ideal Domain are Associates:
 * $c \divides c'$

and:
 * $c' \divides c$

and the proof is complete.