=Allan C. Wechsler extremely reminiscent of a result that was discussed on this forum about a decade ago.
In fact, recalling this, I'd hoped to retrieve some collective memories!
Gosper showed me an absurdly intricate _analytic_ proof of this, and bewailed the lack of a _combinatorial_ proof;
As others have noted, the classic proof is elegant but not intricate--an Euler hack, from around 1740(?). Actually, it's _symbolic_ rather than _analytic_, entailing only formal manipulations of the generating functions. But arguably my own proof "absurdly" generalized Euler by using vectors as exponents (and similar shenanigans). So I wanted to see what saner alternates others would propose.
the inverse operation combines identical factors until none remain.
Alas I kept missing that "combine" must mean "incrementally, pairwise". Otherwise 2.2.2.2.2.2 and 8.8 might both "combine" into 64. Similarly a subtle issue with the old partition query is that effective conversion procedures from one type to another might still be many-to-one. Fortunately JP Grossman's elegant procedures finally clarified this. Very nice! To bring the thread full-circle: You can specialize this general result to roundaboutly prove the old partition identity, which is equivalent to factorizations of prime powers (ie the "exponent vectors" are scalar).