This is not a proof (of money having value per se). Though I was hoping that you would address the issue that I raised in my post (since that would be a proof). I mean, the compliance (or lack thereof) between the notion of money having value as such and the law of diminishing marginal utility...
It is proof. Value is somebody prefering one thing over the other.
We could go on discussing the two types of value again, but this time I wanted to stress that money is value.
So you don't want to address the issue of the discrepancy that I pointed out, okay then. Would money have value to you if you could buy with it only things that you don't need (and couldn't barter with) and, consequently, couldn't that you do? I guess that it wouldn't. Therefore it is not money per se that has value but the value of things that you can buy with it, and only through these things that actually satisfy your needs money obtains its value...
And this pretty much explains the apparent contradiction between the law of diminishing marginal utility and money not conforming to it. If you still disagree, then you have to explain this non-conformance somehow
By this interpretation (which I don't disagree with), cash is effectively representative money with the caveat that cash is not guaranteed to be redeemable for any arbitrary amount, instead relying on market functions. Cash itself is worthless, but it represents X units of "anything and everything" (at least within the issuing country, dependent on their legal tender laws, if any!). Since "whatever we want" is basically the most maximally valuable commodity in the Universe, cash is maximally valuable to everyone (unlike, say, camels or silver nuggets), but only while it represents "whatever we want."
The real issue with the traditional definition of representative currency is that it MUST be legal tender for it to be maximally valuable -- I really don't give half a shit about being able to redeem it for a silver nugget because
1) I have no direct use for silver nuggets and don't want the clutter (there's no single item on Earth I want in such quantity that I'd want to be able to spend my net worth on owning... except maybe Tesla Powerwalls - those things are fucking amazing).
2) in the event of some monetary disaster where I'd actually consider redeeming cash for nuggets, I presume the chance of the nuggets actually being there is very low.
Combined, this makes representative currency (by traditional definition) a giant waste of resources. If the best use we can find for gold, silver, rhodium, and whatever is storing them in a warehouse, aliens should kill us all right now. Similarly, using useful items as a medium of exchange is terrible. -So we use relatively useless items. Cash is fantastic with regards to work efficiency because the paper, fabrics, and polymers we use to make fiat in the world has negligible value (not completely the case with all the semi-effective anti-counterfeiting shit built in these days, but the value's still very low). Who really gives a shit if a bank is storing $10,000,000,000 in nominal value using $291.50 in actual material costs (the actual cost of manufacturing the bills) -- I mean, that's pretty cool that we've been able to add what we were using as money back into the productive economy - but it left us open to policy abuse by the issuers, which turned out to be governments (which is probably far better than banks, at least), so I see bitcoin as the final evolution here and putting the final nail in the coffin of using otherwise-useful resources as something to keep in your pocket or a warehouse. "Redeemable" (if you trust the issuer... a LOT) crypto put a bit of a twist on that, but I don't honestly believe it has any place in the mainstream.