Existence of Greatest Common Divisor/Proof 2

Proof
By definition of greatest common divisor, we aim to show that there exists $c \in \Z_{>0}$ such that:

and:
 * $d \divides a, d \divides b \implies d \divides c$

Consider the set $S$:
 * $S = \set {s \in \Z_0: \exists x, y \in \Z: s = a x + b y}$

$S$ is not empty, because by setting $x = 1$ and $y = 0$ we have at least that $a \in S$.

From the Well-Ordering Principle, there exists a smallest $c \in S$.

So, by definition, we have $c > 0$ is the smallest such that $c = a x + b y$ for some $x, y \in \Z$.

Let $d$ be such that $d \divides a$ and $d \divides b$.

Then from Common Divisor Divides Integer Combination:
 * $d \divides a x + b y$

That is:
 * $d \divides c$

We have that:

Mutatis mutandis:
 * $c \divides b$

Now suppose $c'$ is such that:

and:
 * $d \divides a, d \divides b \implies d \divides c'$

Then we have immediately that:
 * $c' \divides c$

and by the same coin: $c \divides c'$

and so:
 * $c = c'$

demonstrating that $c$ is unique.