Write down a mathematical theorem of your choice and prove it in great detail.
Sure! Here's a theorem and its proof:

Theorem: Every positive integer greater than 1 can be written as a unique product of primes.

Proof:
Let n be a positive integer greater than 1. Then, either n is a prime number or it can be written as a product of two smaller positive integers. If n is a prime number, the theorem is proven. If n can be written as a product of two smaller positive integers, say a and b, where a and b are both greater than 1, then either a or b must be a prime number, because if both a and b were not prime, they would both have a factor greater than 1, which would give us a factorization of n into integers greater than 1 but smaller than n, contradicting the minimality of n's factorization.

This process of factorization can be repeated until all the factors are prime. Hence, we can write n as a unique product of primes.

Q.E.D.