Gemara

(Beraisa): Two witnesses do not join unless they saw the testimony together;

R. Yehoshua ben Korchah says, they join even if they saw the testimony at different times.

(R. Chiya bar Avin): The Halachah follows R. Yehoshua ben Korchah, both regarding (testimony about) land and Metaltelim.

(Ula): The Halachah follows R. Yehoshua ben Korchah regarding land, but not regarding Metaltelim.

Question (Abaye): This implies that there is an argument about land. R. Aba taught that Chachamim agree with R. Yehoshua ben Korchah regarding land!

(Rav Idi bar Avin - Beraisa): Chachamim agree with R. Yehoshua ben Korchah regarding testimony about a Bechor, land, Chazakah, and similarly in a boy or girl (one witness saw two hairs on the front, and one saw two on the back).

Answer: They are Amora'im. R. Chiya bar Aba and Ula argue with them (and say that Chachamim argue about land)!

(Rav Yosef citing Ula): The Halachah follows R. Yehoshua ben Korchah, both regarding land and Metaltelim;

(R. Zeira citing Rav): The Halachah follows R. Yehoshua ben Korchah regarding land, but not regarding Metaltelim.

This is like Rav taught elsewhere.

(Rav): (Testimony of) different admissions join (if 'Reuven admitted to Shimon 'I owe you' in front of Ploni', and later admitted in front of Almoni);

(Testimony of) an admission joins (testimony of one witness who saw) a loan before the admission;

(Testimony of) different loans do not join, nor does an admission join a loan after the admission.

Question (Rav Nachman bar Yitzchak): What is the difference between different loans and different admissions?

Different loans do not join, because they do not testify about the same money. Likewise, different admissions should not join, (perhaps) they are not about the same money!

Answer (Rav Huna brei d'Rav Yehoshua): The case is, when Reuven admitted in front of Almoni, he said 'I already admitted about this debt in front of Ploni'.

Question: Still, Ploni does not know that he testifies about the same money as Almoni does!

Answer: The case is, Reuven later told Ploni 'I admitted in front of Almoni about the debt I admitted to in front of you.'

Question (Rav Huna citing Rava): If so (Reuven must tell each witness that the other witness also knows about this loan), this is just like an admission after a loan! (Why did Rav consider these to be separate cases?!)

(Chachamim of Neharda'a): We join (testimony of) two admissions, two loans, or one of each (in either order).

This is like R. Yehoshua ben Korchah.

Rishonim

Rif and Rosh (Sanhedrin 9a and 3:34): The Halachah follows R. Yehoshua ben Korchah. The testimonies join whether both saw loans or admissions or one saw a loan and the other saw an admission.

Ramah (cited in Rosh, ibid.): When both testify about loans they join only if the lender does not contradict either of them. E.g. Reuven says 'I lent 200 to Shimon; 100 on Sunday, and another 100 on Monday', and he brings one witness for each loan. They join to obligate Shimon 100. But if Reuven says that he lent only 100 and the witnesses testify about different days, they do not join unless we can say that both saw the same loan that one of them erred. E.g. if one says that it was on the second of the month and the other says the third; one of them erred about which day was Rosh Chodesh. If not, one of them is lying, so at most one testifies truthfully. Shimon takes a Shevu'ah mid'Oraisa to contradict the testimony, and he is exempt.

Rosh: Seemingly there is no Shevu'ah mid'Oraisa, because the lender says that one of them testified falsely. When two witnesses contradict each other, they are Pesulim for that testimony. However, that is only when they argue about the same event, e.g. one says that Ploni murdered and the other says that he did not (at that time). Here, they testify about different matters, and the lender says that one of them lies. The other was not contradicted, so he obligates a Shevu'ah mid'Oraisa.

Nimukei Yosef (ibid.): Likewise, if David testifies about a loan and Levi testifies about an admission at an earlier time, they join only if Reuven says that there were two loans. If David testifies that Shimon owed 200, Levi testifies that he owed 300, Moshe testifies that he owed 400 and Shmuel testifies that he owed 500, we join the testimonies of 200 and 300 to obligate 200, and we join the testimonies of 400 and 500 to obligate another 400.

Rambam (Hilchos Edus 4:2,3): In monetary cases, even if the witnesses saw one after the other, their testimony joins. If David says 'I saw Shimon borrow from or admit (that he owes) to Reuven on Sunday', and Levi says 'I saw Shimon borrow from or admit to Reuven on Monday', their testimony joins.

Rambam (3): The same applies if one saw a loan and the other saw an admission.

Poskim

Shulchan Aruch (CM 30:6): In monetary cases, even if the witnesses saw one after the other their testimony joins. If David says 'I saw Shimon borrow from or admit (that he owes) to Reuven on Sunday', and Levi says 'I saw Shimon borrow from or admit to Reuven on Monday', their testimony joins. The same applies if one of them saw a loan and the other saw an admission.

Beis Yosef (DH Ein): Chachamim of Neharda'a conclude that the Halachah follows R. Yehoshua ben Korchah in every case, i.e. whether the first or second or both witnesses saw a loan or admission.

Shulchan Aruch (7): When both testify about loans they join only if Reuven says that he lent at the two times. But if Reuven says that he lent only 100 and the witnesses testify about different days, they do not join unless we can say that both saw the same loan and one of them erred. E.g. one says that it was on the second of the month and the other says the third; they erred about which day was Rosh Chodesh, so they join. If they surely testify about different loans, surely one of them is lying. Shimon takes a Shevu'ah mid'Oraisa and is exempt.

Beis Yosef (DH v'Chosav): When the latter testimony is an admission, witnesses who saw at different times always join, for perhaps he admitted to the same loan as the first testimony. Also (testimony of) a loan after an admission is Kosher, for perhaps the borrower was soliciting a loan and admitted before he received it. However, if both testify about loans at different times and the lender says that he lent only once, surely one witness lies.

Beis Yosef (CM 30 DH Achen Rabbeinu): R. Yerucham stipulates that the lender does not contradict the witnesses. He does not distinguish whether the first or last or both were loans or admissions. However, I proved that we are concerned only when both saw loans! Perhaps even R. Yerucham refers only to this case.

SMA (21): The Rosh and Nimukei Yosef are concerned only for a loan after a loan.