I have this theorem to prove:
Every integer n > 1 is either prime or can be expressed as a product of primes.
I want to know if my proof is sound and if not, in what way?
My attempt:
Note: Negation of XOR is the biconditional
Theorem in other words:
If n is an element of integers and n > 1, then n is prime XOR n can be expressed as a product of primes.
Proof by contradiction:
Assume: If n is an element of integers and n > 1, then n is prime <=> n can be expressed as a product of integers.
I try to prove, n is prime <=> n can be expressed as a product of primes, is always false as follows:
n is prime =>n is divisible by 1 and itself only =>n cannot be expressed as a product of primes.
n can be expressed as a product of primes =>n is not divisible only by 1 and itself. =>n is not prime.
Therefore the bi-conditional statement is always false, hence we have a contradiction.
Thus the theorem holds.
Q.E.D
What you ask follows from the Fundamental Theorem of Arithmetic, which states that every positive integer great than $1$ has a unique prime factorization (up to arrangement of the prime factors).
Hence, it follows that every integer $n > 1$ is either prime or a product of primes.