P(n): All natural numbers greater than 1 can be expressed as a unique product of primes where order doesn't matter.
By strong induction:
Base Case n=2: 2=2*1 so base case holds as this is a unique prime representation.
Inductive step: Assume, for purposes of induction, that P(n) holds for all natural numbers up to n i.e. 2,3,4...n.
Now take the number n+1.
If n+1 is prime, it has a unique representation (by definition of being prime) and we are done.
If it is composite, it can be written as a product of natural numbers (not necessarily unique or prime) all of which are smaller than n+1. - but we know from IHOP that all numbers up to n have unique prime representations - so we can replace all the numbers forming n+1 by their unique representations (as given by IHOP) - hence, all possible representations of n+1 will reduce to the same unique product of primes after substituting.
Hence, by principle of mathematical induction, the P(n) holds for all natural numbers.