In LN, how could a payer prove to 3rd party that he indeedly made a payment to the payee?
In lightning the receiver of the money comes up with a random secret and comits to it via the payment hash. Once all the hashed time lock contracts (htlcs) are set up along a path through the network from the sender to the receiver the recipient will release the secret preimage for the chain of htlcs to settle. The preimage can be seen as a proof of payment. However as with the onchain case there is some doubt. Every node along the route who participated in the payment will know the primage of the payment hash and could thus argue that they initiate the payment