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1) MISHNAH: SUKAH YESHANAH

(a) (Beis Shamai) A previously built Sukah is Pesulah.
(b) (Beis Hillel) It is Kesheirah.
(c) A Sukah is Yeshanah if built 30 days before Sukos.
(d) It is Kesheirah, however, if it was built LeShem Sukah.

(a) Question: What is Beis Shamai's rationale?
(b) Answer: The Pasuk (Vayikra 23:34) indicates that the Chag and the Sukah must be LeShem Mitzvas HaShem.
(c) Question: How will Beis Hillel interpret that Pasuk?
(d) Answer: It teaches the Din of R. Sheshes citing R. Akiva (explained by R. Yehudah b. Beseira) that it is forbidden to make personal use of the Sukah boards for seven days.
(e) Question: The Pasuk must teach this Din according to Beis Shamai, as well (so how can it also teach Lishmah)!?
(f) Answer: Indeed, Beis Shamai interpret Devarim 16:13 to teach that the Sukah must be made Lishmah.
(g) Question: How will Beis Hillel understand that Pasuk?
(h) Answer: It permits constructing a Sukah during Chol HaMoed (unlike Beis Shamai who prohibit making a Sukah for less than all seven days).
(i) Question: Do Beis Hillel (who do not require Lishmah) disagree with R. Yehudah citing Rav that Tzitzis need to be made Lishmah (and, presumably, fashioning all Mitzvos)?

1. R. Yehudah cited Rav as teaching that Tzitzis must be affixed to the garment Lishmah, and not be made from residual threads from the weaving of the garment.
2. While Shmuel argues with Rav (as pointed out when R. Yehudah cited this Din before him) regarding the actual spinning of the Tzitzis threads Lishmah, the issue of affixing them Lishmah is not under dispute!

(j) Answer: Tzitzis is unique, due to Lecha (in Devarim 22:12).
(k) Question: But Lecha appears by Sukah, as well (ibid. 16:13)?
(l) Answer: Lecha there prohibits the use of a stolen Sukah.
(m) Question: Then Lecha should teach the same by Tzitzis (not the Din of Lishmah)!?
(n) Answer: The word LaHem (in Bamidbar 15:38) already taught Gezulah by Tzitzis (freeing Lecha to teach Lishmah).

3) MISHNAH: DIFFERENTIATING A ROOF FROM S'CHACH

(a) A Sukah built under a tree is (Pesulah) as though it were built indoors.
(b) (Tana Kama) If a Sukah were built above another, only the upper one is Kesheirah.
(c) (R. Yehudah) If the roof/floor between them is not fit for residence, then the lower Sukah is Kesheirah.

(a) (Rava) This Pesul only applies if the shade of the tree is greater than its light.
(b) Question: Whence do we know this?
(c) Answer: From the Mishnah's comparing it to a Sukah built indoors (and not simply stating that it is Pesulah), it must share this quality with the indoors (shade more than light).
(d) Question: But even if the density is less, the tree still creates a mixture of S'chach Pasul with Kasher (since even the limited density of the tree contributes to the S'chach below, and should thus invalidate it)!?
(e) Answer (R. Papa): He lowered an mixed the tree's branches with the Kosher S'chach, such that it is Batel b'Rov.
(f) Question: If Rava was speaking where he had mixed the S'chach, then it is *obvious* that it is not Pasul!?
(g) Answer: We might have thought to prohibit a case of mixed S'chach lest he come to permit such S'chach even when not mixed.
(h) Question: But the fact that we do not make such a Gezeirah is clear from the later Mishnah, which permits using Gefen, etc. for S'chach, provided that the Kosher S'chach is in the majority, and, by implication, it is mixed!?
(i) Answer: We might, from that Mishnah, inferred that the use of such S'chach is only b'di'Eved (an impression which our Mishnah thus comes to correct).

(a) The Beraisa derives the prohibition of one Sukah above another from Vayikra 23:42 (the singular spelling [Kesiv] of Sukos teaches that a Sukah may have only one S'chach).
(b) Question: But the plural word (Sukos as read [K'ri]) indicates the opposite!?
(c) Answer (R. Nachman b. Yitzhok): We must take into account the singular form in which it is written.
(d) R. Yirmiyah teaches that there are illustrations where:

1. Both the upper and lower Sukos are Kesheiros.
2. Both are Pesulos.
3. The lower is Kesheirah and the upper is Pesulah.
4. The lower is Pesulah and the upper is Kesheirah.

(e) The Gemara gives an illustration for each of the cases of R. Yirmiyah, supplying the details of which Sukah had adequate S'chach, and which had inadequate S'chach; likewise which S'chach was within 20 Amos and which was above
(f) Question: But these are *obvious* applications of known Halachos!?
(g) Answer: The cases are justified because we might have prohibited a Sukah which R. Yirmiyah permits.

1. The case wherein the lower S'chach was Kesheirah (more shade than light), and the upper was Pesulah (less shade than light, within 20 Amos) is acceptable.
2. We might have thought to prohibit it given that if the upper Sukah were above 20 Amos, its S'chach is Pesulah, based on its height, regardless of the fact that it is here Pesulah owing to its sparsity.
3. If the lower Sukah relies on the sparse S'chach of such an upper S'chach, it would be combining Kosher and Pasul S'chach, thus invalidating the lower Sukah.
4. A case where the upper S'chach is similarly Pesulah owing to its sparsity could, perhaps, be deemed Pasul lest it be confused with the case where the upper S'chach is above 20.
5. R. Yirmiyah is thus teaching that we do not impose such a Gezeirah.