Rami bar Chama draws an inference from the Mishna’s statement that when two people pick up a lost object together, they both acquire it. From this we see that each one intended to acquire it not just for himself, but also for the other person who is picking it up with him.
The question is, what is the inference? Perhaps they are both picking up each half only for themselves, and they each acquire the part that they have picked up, by the power of their own act of acquisition?
נ”ל בע”ה that if they indeed had the intention to acquire the object only for themselves, that would mean that they were each only trying to acquire the entire object. Thus, being that they picked it up together, neither of them would have acquired it, because if two people try to completely acquire the same thing at the same time, their acts cancel each other out, and it remains ownerless.
Therefore, if we say that they have both acquired half of it, it must be that they were acquiring it in a way that each would have half. The intent of each is necessary on behalf of the other party, otherwise their selfish intent would block each other’s acquisition attempt. Thus, we see that each one is helping the other acquire his half, and therefore Rami bar Chama proves that a person has the ability to acquire something for someone else.
This answer, however, requires further thought, because if it is correct, then it should not be necessary to have active thoughts to help the other party acquire, rather it should be enough for him to passively permit the other party to acquire his side.
On further thought, however, we already have seen that when they split up the object, it is not per se split based on exactly which side or which part is being held, but rather, it can be split in alternative ways (e.g. the talis with gold). Clearly, each person is helping the other party to acquire the garment in its entirety, albeit as a partner. This is the function that allows each party to request an alternate splitting up of the garment (or object).
It therefore becomes clear that each party is helping the other to acquire the entire object as a partner, and this can only be done through זכייה – actually acquiring it for the other party. This is Rami bar Chama’s proof.