Euclidean Algorithm/Proof 2

Proof
Suppose $a, b \in \Z$ and $a \lor b \ne 0$.

Let $d = \gcd \set {a, b}$.

By definition of common divisor:
 * $d \divides a$

and:
 * $d \divides b$

Hence from Common Divisor Divides Integer Combination:
 * $d \divides \paren {a - q b}$

That is:
 * $d \divides r$

Thus $d$ is a common divisor of $b$ and $r$.

Let $c$ be an arbitrary common divisor of $b$ and $r$.

Then:
 * $c \divides \paren {q b + r}$

That is:
 * $c \divides a$

Thus $c$ is a common divisor of $a$ and $b$.

Hence by definition of GCD:
 * $c \le d$

Hence, again by definition of GCD: $d = \gcd \set {b, r}$

Then we work down the system of equations: