Fredrick: That's the same counterexample I settled on. Amusingly, my "pick a random set of numbers" approach kept finding ones that worked! (2|4, 3|6 => 5 | 10, for example).

This is one demonstration of why just finding a few small examples to satisfy ourselves an assertion is true isn't sufficient. It's easy to mislead yourself into thinking you've stumbled upon a relation that doesn't actually exist.