# 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 $a = -b$.

## Proof

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

Then by definition of divisor:

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

Thus:

 $\displaystyle a c d$ $=$ $\displaystyle a$ $\displaystyle \leadsto \ \$ $\displaystyle c d$ $=$ $\displaystyle 1$ $\displaystyle \leadsto \ \$ $\displaystyle d$ $=$ $\displaystyle \pm 1$ Divisors of One $\displaystyle \leadsto \ \$ $\displaystyle a$ $=$ $\displaystyle \pm b$ as $a = b d$

$\blacksquare$