Existence of Greatest Common Divisor/Proof 4

Proof
From the Euclidean Algorithm, we have calculated a sequence $\tuple {r_1, r_2, \ldots r_{n - 2}, r_{n - 1}, r_n}$ such that:
 * $b > r_1 > r_2 > \dotsb > r_{n - 2} > r_{n - 1} > r_n = 0$

We have that:
 * $r_{n - 1} \divides a$

and:
 * $r_{n - 1} \divides b$

Working backwards from the final equation, we see that:
 * $r_k \divides r{k - 1}$

for all $k$ such that $1 < k \le n$.

Hence, if $d \divides a$ and $d \divides b$, we can use induction to proceed through the Euclidean Algorithm and see that $d$ divides $r_1, r_2, \ldots, r_{n - 2}, r_{n - 1}$.

Thus we see that $r_{n - 1}$ fulfils the criteria to be the greatest common divisor of $a$ and $b$.

Suppose $c_1$ and $c_2$ are both greatest common divisors of $a$ and $b$.

Then by definition there exist $g, h \in \Z_{>0}$ such that:
 * $g c_1 = c_2$
 * $h c_2 = c_1$

Hence:
 * $c_2 = g h c_2$

and so:
 * $g h = 1$

That is:
 * $g = h = 1$

and so:
 * $c_1 = c_2$

That is, the greatest common divisor of $a$ and $b$ is unique.