Square of Coprime Number is Coprime

Theorem
Let $a$ and $b$ be coprime integers:
 * $a, b \in \Z: a \perp b$

Then:
 * $a^2 \perp b$


 * If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.

Proof
Let $a \perp b$.

Let $a^2 = c$.

Let $d = a$.

As $a \perp b$ it follows that $d \perp b$.

From :
 * $a d \perp b$

But $a d = c = a^2$.

Hence the result.