That the Messenger of Allah (ﷺ) said: "Whoever kills someone in battle, having a proof for that, then his goods are his."

There is a story with this Hadith.
Jami` at-Tirmidhi Book 21, Hadith 20
← Previous Hadith "It has not reached me that the Messenger of Allah (ﷺ) gave the Nafl during every expedition, but it has been conveyed to me that he gave the Nafl in some of them. That is according to the discretion of the Imam during the beginning of the division of the spoils or the end of it."

Ibn Mansur said: "I said to Ahmad: 'The Prophet (ﷺ) gave the Nafl when he divided the fourth, after the Khumus, and when he was returning (he gave) the third from the Khumus.' So he said: 'The Khumus is taken, and then the Nafl is given from what remains, nothing beyond this.''"

This Hadith is (understood) as Ibn Musayyab said: "The Nafl is from te Khumus." Ishaq said as he said.

Jami` at-Tirmidhi Book 21, Hadith 19 Next Hadith → "The Nafl is when the Imam says: 'Whoever got something, then it is his. And whoever killed a fighter, then his goods are his.' So it is allowed, and there is no Khumus taken from it." Ishaq said: "The goods are for the one who did the killing, unless it is something that is a large amount." So he saw that the Imam could take the Khumus from that, just as 'Umar bin Al-Khattab did.

Jami` at-Tirmidhi Book 21, Hadith 21