1) THE USE OF AIRSPACE
QUESTION: The Gemara discusses a case of a pit in an alleyway between the walls of two Chatzeros, where the pit is at least four Tefachim away from each wall. According to Shmuel, it is forbidden for the residents of both Chatzeros to use the pit on Shabbos, because one Chatzer's usage of the airspace of an area prohibits the other Chatzer from using it ("Adam Oser Al Chaveiro Derech Avir"). Since they share the usage of that airspace, neither one may use it. According to Rav, if the Chatzeros use only the airspace, then it is not considered a "usage" and both Chatzeros may use it on Shabbos.
The RASHBA asks that we find cases in which both Chatzeros can use a particular domain only by lifting something onto it (76b, 83b), such as the case of "la'Zeh b'Zerikah v'la'Zeh b'Zerikah." In those cases, all agree that the domain is forbidden to be used because it is a shared domain and each Chatzer has equal rights to use it. Why is this case different?
(a) Most Rishonim answer that the use of an area through Zerikah is considered a valid form of usage, even though it is not an easy one. Since each Chatzer is able to use the area with a valid usage, it prohibits the other from using it on Shabbos. However, if the only way to use the area is by reaching across four Tefachim, that is not considered a valid usage, and it is as if neither Chatzer is able to use the area at all. (See RITVA.)
(b) The RASHBA adds that this case is not comparable to the case of "Zeh v'Zeh b'Zerikah," because the usage of the domain between the two Chatzeros is not a consistent, regular usage. Since the domain is part of neither Chatzer, and neither Chatzer uses it on a regular basis, whatever usage each Chatzer has from that area cannot prohibit it.
2) THE PROOF THAT SHMUEL HOLDS THAT USE OF AIRSPACE IS SIGNIFICANT
QUESTION: The Gemara attempts to prove that Shmuel maintains that one person's (or Chatzer's) right to use the airspace of an area prohibits the other person (or Chatzer) from using it on Shabbos ("Adam Oser Al Chaveiro Derech Avir"). The Gemara cites a ruling of Shmuel regarding a house in a Chatzer which has a roof that is adjacent to Reshus ha'Rabim. Since the people in Reshus ha'Rabim use the roof, the residents of the Chatzer (on the other side of the house) may use the roof only if a permanent ladder leads from the Chatzer to the roof. Such a ladder shows that they have established the exclusive right to use the roof.
Why is it necessary for there to be a permanent ladder in order to permit the Chatzer to use the roof? Why would the Chatzer not be permitted to use the roof without such a ladder? It must that the people in Reshus ha'Rabim are able to prohibit the Chatzer from using the roof even though they use only the airspace of the roof. We see that Shmuel considers use of the airspace to be a significant form of use.
What is the Gemara's proof? If the people in Reshus ha'Rabim are able to prohibit the Chatzer from using the roof merely by using its airspace, then the residents of the Chatzer should need only a temporary ladder, and not a permanent one, to permit them to use the roof. The Gemara (84b) says that a ladder placed between the Chatzer and the roof temporarily makes the area into one that is easily used (and this suffices to establish for the Chatzer the exclusive right to use the area, since the Reshus ha'Rabim's use of the roof is only with difficulty.) It must be that there is another reason for why a permanent ladder is necessary, and it is not because of the Reshus ha'Rabim's use of the airspace! Rather, a permanent ladder is necessary because the people in Reshus ha'Rabim have full and easily accessible usage of the roof (as the Gemara explains, they place their hats there), and that is why the Chatzer must lean a permanent ladder against the roof in order to "reclaim" it and prevent Reshus ha'Rabim from using it. Why, then, does the Gemara attempt to prove from here that use of the airspace prohibits the others from using the area?
ANSWER: The Gemara initially assumed that the roof is within ten Tefachim of Reshus ha'Rabim, as the Gemara proposed at one point earlier (84b, in the Havah Amina). Therefore, the Reshus ha'Rabim's use of the airspace indeed is an easy use, and the Chatzer needs to counter it with a permanent ladder.