Divisor of One of Coprime Numbers is Coprime to Other/Proof 1

Proof
Let $a \perp b$ and $c > 1: c \mathop \backslash a$.

Suppose $c \not \perp b$.

So by definition of not coprime:


 * $\exists d > 1: d \mathop \backslash c, d \mathop \backslash b$.

But from Divisor Relation is Transitive:
 * $d \mathop \backslash c, c \mathop \backslash a \implies d \mathop \backslash a$

So $d$ is a common divisor of both $a$ and $b$.

So, by definition, $a$ and $b$ are not coprime.

The result follows by Proof by Contradiction.