Divisor Relation is Antisymmetric/Corollary/Proof 2

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Corollary to Divisor Relation is Antisymmetric

Let $a, b \in \Z$.

If $a \divides b$ and $b \divides a$ then $a = b$ or $a = -b$.


Let $a \divides b$ and $b \divides a$.

Then by definition of divisor:

$\exists c, d \in \Z: a c = b, b d = a$


\(\ds a c d\) \(=\) \(\ds a\)
\(\ds \leadsto \ \ \) \(\ds c d\) \(=\) \(\ds 1\)
\(\ds \leadsto \ \ \) \(\ds d\) \(=\) \(\ds \pm 1\) Divisors of One
\(\ds \leadsto \ \ \) \(\ds a\) \(=\) \(\ds \pm b\) as $a = b d$