Explanation: We should say the first half of the day they pray and request mercy, and the other half they investigate matters of the city;

The Gemara answers that this cannot be...

Citation (Yerushalmi): On Tish'ah b'Av and Yom Kipur, Yosef brei d'R. Yehoshua ben Levi would wash his hands, clean them on a cloth and pass it over his eyes;

Citation (ibid.): R. Yonah soaked a rag and put it under Kar'ei, i.e. he put a rag under his pillow in order to dry it, and passed it over his eyes.

Opinion #1 (Avi ha'Ezri): Now that there is no Ta'anis Tzibur in Bavel except for Tish'ah b'Av, it is permitted even in warm water, even the 17th of Tamuz and Tzom Gedalyah;

Opinion #2 (R. Yo'el): R. Yo'el, his father, forbade bathing in warm water.

Explanation: They are [bathhouses] of hot water.

Explanation: I.e. if it were forbidden in cold water, would we teach that we dam the rivers?! Obviously we cannot!

I.e. similarly, do not infer from here that it taught that we lock the bathhouses [that it is forbidden only in hot water].

Explanation: They are Mayim She'uvim (water that was in a Kli, which is Pasul for a Mikveh.)

Observation: We did not say that we find Tevilah Kosher in hot water, e.g. they were heated and Humshach (poured onto the ground in order to flow) into a pond. We said that we discuss hot springs of Teverya;

Inference: All She'uvim, even through Hamshachah, do not help for Tevilah.

Implied question: We say that totally [through] Hamshachah is Kosher (Temurah 12b)!

Answer: This means that it does not disqualify a Mikveh like other Mayim She'uvim. However, surely one may not immerse [if the entire Mikveh was through Hamshachah].

Explanation: We say that [bathing] is permitted in cold water. [We suggested that] also an Avel is permitted in cold water. This is difficult for Rav Chisda, who forbids an Avel in hot and cold water;

We answer that this refers to other laws, i.e. wearing shoes and anointing. It does not apply to cold water.

Explanation: [He may lighten his hair with a razor] in the way of cutting, but not in the normal way, i.e. shaving;

Proof: It says that he may lighten his hair, but it did not say that he may shave.

Explanation (Ri): This does not literally mean "she is not permitted", rather, she is not obligated;

Inference: A Na'arah is obligated.

Explanation: It says that a Na'arah is Resh'ah (obligated to make herself repulsive,) i.e. forbidden [to bathe]. How can you say that cold water is permitted?!

This is difficult for Rava, who permits in cold water. We answer that the Beraisa teaches about painting eyes and braiding hair.

Explanation: The Tana taught about doing Melachah, and omitted 24 Berachos;

I.e. on a Ta'anis there are 24 Berachos, for they add six Berachos to Shemoneh Esre, like it says below (15a).

Explanation: If there is only one matter omitted, the Tana should not have omitted it.

Question: Here we say that the Tana did not omit things, for it says "Ein Bein" (the only differences between...). This implies that these are the only differences;

In Megilah (8a), it was taught "Ein Bein Nedarim and Nedavos, except that one has Acharayus for Nedarim (he must bring another Korban if the one he separated became lost, stolen or blemished), but there is no Acharayus for Nedavos";

We know that there are other differences! Nedarim may be brought only from Chulin (but not from Ma'aser), for any Chiyuv comes only from Chulin!

Nedavos come from Ma'aser, like it says "v'Zavachta Shelamim v'Achalta", which is written regarding Ma'aser.

Note: The Rashash (Megilah 8a Tosfos DH Ein) points out that the verse does not discuss Ma'aser. It discusses Har Eval! However, we learn a Gezeirah Shavah between Shelamim and Ma'aser (Menachos 82a). The Gemara there says that Ma'aser is called Shelamim. Rashi Kesav Yad says that we learn from the Gezerah Shavah.

Answer: Also there, both are the same. [Only] regarding Shelamim there is a difference. "Ein Bein" was taught regarding Olos. The only difference between Olos Nedarim and Olos Nedavos is that one has Acharayus for Nedarim, but not for Nedavos;

This is the only difference between Olos Nedarim and Olos Nedavos. All Olos are only from Chulin!

Explanation: We explain below (15b) that we take the Teivah to the city square in the last fasts;

[The Gemara asks] "do you really think [that there are no other differences between the middle fasts and the last ones]? Also the Teivah was omitted!

Explanation: You cannot say that the Tana discusses only Isur, for [he mentions] blowing the Shofar and locking stores. They do not entail any Isur!