[{'from': 'human', 'value': 'Write down a mathematical theorem of your choice and prove it in great detail.'}, {'from': 'gpt', 'value': "Sure! Here's a theorem and its proof:\n\nTheorem: Every positive integer greater than 1 can be written as a unique product of primes.\n\nProof:\nLet 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.\n\nThis process of factorization can be repeated until all the factors are prime. Hence, we can write n as a unique product of primes.\n\nQ.E.D."}]