Product of Coprime Pairs is Coprime

Theorem
Let $a, b, c, d$ be integers.

Let:
 * $a \perp c, b \perp c, a \perp d, b \perp d$

where $a \perp c$ denotes that $a$ and $c$ are coprime.

Then:
 * $a b \perp c d$


 * If two numbers be prime to two numbers, both to each, their products also will be prime to one another.

Proof
Let $e = a b, f = c d$.