> how do you prove that you can spend the received funds?

You prove it by producing a partial signature for the transaction (a transaction essentially requires a multisig by both sender and receiver).

> And if I accidentally send it twice

You couldn't send it twice because the transaction consumes an input utxo. Paying twice would require two different transactions with two different multisignatures.

> You prove it by producing a partial signature for the transaction (a transaction essentially requires a multisig by both sender and receiver).

Oh, ok, I misunderstood what this meant.

> You couldn't send it twice because the transaction consumes an input utxo. Paying twice would require two different transactions with two different multisignatures.

This assumes the receiver is trust-worthy, that they counter-sign the transaction if it was accidental. So if I accidentally pay twice for a single item, the seller can just sign both transactions and send me only one item, and I can't do much about it. Of course, they can also send 0 items, which is why blockchains can't secure transactions in a useful sense - you need the law and insurance and other social mechanisms for that.