In Gemara Berachos, on Daf mem-hei, amud rishon, the commentary of Tosfos begins with the opening phrase, "three who ate together are not permitted to separate." The discussion explores a dispute regarding whether two people who ate together are required to perform a formal invitation to Grace After Meals. One opinion holds that two people do not have this obligation. To support this, the text brings a proof from the Mishna, which states that three who ate together are not permitted to separate. The logic of this proof is that the restriction applies specifically to a group of three, implying that the obligation does not exist for only two.
Perush Rashi explains the reasoning behind this proof. If one were to assume that two people could indeed perform a formal invitation, why would one person be forbidden from leaving a group of three? If he left, the remaining two could simply perform the ritual themselves without losing the blessing. Therefore, the fact that he is not permitted to go away and leave the others behind serves as a proof that the remaining two would lose the opportunity for a formal invitation entirely. According to this view, the prohibition exists to prevent the loss of the ritual for the group.
However, Tosfos finds this proof difficult to accept. It is possible that the reason a person is not permitted to separate is not because the remaining two would lose the ritual, but because the individual who leaves would be exempting himself from a commandment. He should not be allowed to remove the obligation of the formal invitation from himself. If this is the case, the restriction on a group of three does not necessarily prove that two people cannot perform the ritual; it might only mean that once a group of three is formed, no one should forfeit their personal obligation. Therefore, the initial proof remains unclear.
To resolve this, Tosfos suggests that the primary challenge actually relies on the latter part of the Mishna, which teaches that four people are also not permitted to separate. In a case of four people, if they were to split into two groups of two, and if two people were capable of performing a formal invitation, there should be no problem with them separating. The fact that they are forbidden from doing so proves that a group of two does not have the obligation. Although the text quotes the first part of the Mishna, it implies the entire passage, including the latter part. This is a common stylistic occurrence where the beginning of a text is cited as a shorthand for the relevant logic found later in the passage.
Tosfos then considers a different approach, questioning why the text would not simply cite the latter part of the Mishna explicitly if that were the source of the proof. Therefore, Rabbeinu Yehuda explains that the proof is indeed derived from the first part of the Mishna. He argues that the restriction is not focused on the individual. If the concern were only that the individual should not lose his own obligation, the Mishna should have stated that the individual is not permitted to separate or that one of them is not permitted to separate. Instead, the Mishna uses the plural form, stating they are not permitted to separate. This phrasing implies that if one person leaves, the obligation is abolished for everyone in the group. This confirms that the presence of the third person is essential for the ritual to exist at all, proving that two people alone do not have the obligation for a formal invitation.