Natural Number is Prime or has Prime Factor

Theorem
Let $a$ be a natural number greater than $1$.

Then either:
 * $a$ is a prime number

or:
 * there exists a prime number $p \ne a$ such that $p \mathop \backslash a$

where $\backslash$ means is a divisor of.

Proof
By definition of composite number $a$ is either prime or composite.

Let $a$ be prime.

then the statement of the result is fulfilled.

Let $a$ be composite.

Then by :
 * $\exists p: p \mathop \backslash a$

where $p$ is a prime number.

The result follows by Proof by Cases.