Existence of Greatest Common Divisor/Proof 3
Let $a, b \in \Z$ be integers such that $a \ne 0$ or $b \ne 0$.
Then the greatest common divisor of $a$ and $b$ exists.
From Elements of Euclidean Domain have Greatest Common Divisor, $a$ and $b$ have a greatest common divisor $c$.
This proves existence.
Suppose $c$ and $c'$ are both greatest common divisors of $a$ and $b$.
- $c \divides c'$
- $c' \divides c$
and the proof is complete.