Divisor of One of Coprime Numbers is Coprime to Other

Theorem
Let $a, b \in \N$ be numbers such that $a$ and $b$ are coprime:
 * $a \perp b$

Let $c > 1$ be a divisor of $a$:
 * $c \mathop \backslash a$

Then $c$ and $b$ are coprime:
 * $c \perp b$