We are moving on to Bava Metzia, Chapter 1, Mishnah 2, which is a direct continuation of the previous Mishnah.
The text of the Mishnah:
"Hayu shenayim rochevin al gabei behemah" - two people sit together on the back of an animal, such as a horse.
"O shehayah echad rochev ve'echad manhig" - one sits on its back and the other causes it to walk, whether he leads it by the reins or strikes it with a stick.
"Zeh omer kulah sheli vezeh omer kulah sheli" - each one claims that he found it, and therefore the ownership of the entire animal belongs to him. The exact details of the action make no practical difference: whether they both climbed on it at the same time, or whether one claims he climbed on it and the other claims he led it.
And the law, as we saw in the previous Mishnah: "Zeh yishava she'ein lo bah pachot mechetzyah, vezeh yishava she'ein lo bah pachot mechetzyah, veyachaloku". Each of them swears that he does not have less than half of it - and as we established in the previous Mishnah, the meaning of this phrasing is that he has a share in the animal, and his share is not less than half - and after the oath they divide it between them.
How the division is carried out in practice:
It is possible that the division here is identical to the division in the previous Mishnah, and it is possible that it is not identical. When saying "they divide it," the rule - both for a cloak and for an animal - is that if the object can be physically divided in two without its value decreasing as a result of the division, we divide it in practice and give half to each one. This is the law, for example, with a gold bar or a barrel of wine, where the substance itself can be divided half and half. But for something that will be destroyed in the division, such as a coat or the cloak in the first Mishnah, and certainly an animal - we do not cut it in two. Rather, they sell the cloak or the animal, and they divide the monetary proceeds half and half.
What is the novelty of the Mishnah?
Seemingly, one should ask why this Mishnah was needed at all, since so far nothing has been said in it other than that two people fighting over an animal, both having an equal share in it, divide it with an oath - and this is exactly the law of the previous Mishnah. Rather, our Mishnah comes to add one point: that riding on the back of the animal is considered an independent act of acquisition in and of itself.
And the Rishonim argue about this:
Riding on the animal is a separate and independent type of acquisition.
Riding is merely a method of performing pulling - the accepted way to acquire an animal - since the rider causes it to move by sitting on its back. And it is possible that even if he does not cause it to move, the very act of sitting on its back is considered pulling.
In any event, the point is that riding in and of itself is a method of acquisition. And practically, as the text of the Mishnah states, there is no difference whether he rides in the front or in the back - both acquire it equally when two sit on its back; and similarly when one pulls the animal with a strap or a stick and the other rides on it, neither has an advantage over his friend.
"Bizman shehem modim o sheyesh lahen edim":
When the two individuals arguing over the garment or the animal admit that they both lifted it together and acquired it together, there is no need for an oath and they divide it between themselves. The same law applies when there are witnesses. The Meiri explains that the witnesses testify that they both lifted it as one, while the Rashash explains that each of them has witnesses testifying that he lifted it first - and both paths lead to the same conclusion: "cholkin belo shevuah" - they divide it without an oath, and there is no longer a need for any oath.
Seemingly, this law is self-evident, and one must ask what it comes to add: if they admit that they are going to divide it between themselves, why would we even consider that they should need to swear? The answer is that the Mishnah is teaching a completely different law - that when one lifts a found object on behalf of his friend, his friend acquires it. The classic case is a person who sees an object lying on the ground, thinks that his friend would want it, and lifts it for him - and the object becomes the friend's property.
And here, since both of them are each lifting half a garment: if we were to say that a person cannot lift a found object on behalf of his friend, it would turn out that each of them only lifted half a garment from the ground, which is not a proper act of acquisition, and the garment would not be acquired by either of them. The Mishnah therefore teaches us that even a joint lifting is effective: each one lifts his part, and that is sufficient for them to acquire it jointly. And in the case of our Mishnah, we view them as having acquired the object together, and therefore they divide it half and half.