Explanation: "Hifginu" is an expression of lamenting and screaming;

If the text says "Shamayim", it refers to Hash-m. They were screaming to Hash-m "why should the Nochrim say 'where is your G-d'"?, and to the Nochrim they said "are we not your brothers?!"

Explanation: They added a day to Adar because the moon was not seen until day 30, and Rosh Chodesh Nisan was day 31.

Therefore, we cannot forbid the 30th due to being the day after the 28th, for the 29th interrupts between them;

Therefore, They needed to forbid it (the 30th) due to being the day before Rosh Chodesh Nisan.

Explanation: If any decree will occur from Rosh Chodesh... the eighth itself will be forbidden, for it is [when Shavu'os was established].

Question (Rashi): Still, we could have taught "from the ninth...", for even if due to a decree, we nullify the first seven days, the eighth will be forbidden, for it is the day before when Shavu'os was established!

Answer (Rashi): If the Yom Tov itself (the eighth is one of the days of the Yom Tov of establishment) of the Tamid was Batel, we will not decree to forbid it due to the day before when Shavu'os was established!

Support: We answer like this below regarding the day of Turainus.

Question: Since he said that the Halachah follows R. Yosi, why do we fast on the day before Purim?

Suggestion: It is because Megilas Ta'anis was Batel.

Rejection: It was not Batel regarding Chanukah and Purim!

Answer #1: We say that Chanukah and Purim were not Batel, i.e. their laws, to make them a Yom Tov and not fast on them. Regarding laws of before them, they were Batel.

Answer #2: Since the days before Purim were Batel from their Yom Tov that they used to make on them, and their Yom Tov was Batel, we should say that they are Batel also from the law of the day before Purim.

Note: Seemingly, the question was only regarding the 13th (Yom Nikanor). Why does Tosfos discuss the days (plural) before Purim? Indeed, in our text of Tosfos, sometimes Tosfos mentions the day before Purim (and Keren Orah connotes that his question was only about the 13th), but the Bach changes the references to be plural, except for the very end of Tosfos. Perhaps Tosfos refers also to the 12th (Yom Turainus). Why do we fast on the 12th, e.g. when the 14th is on Shabbos and we read the Megilah [in unwalled cities] on the 13th, and it is considered like Purim in every respect? Alternatively, Tosfos holds that in a leap year, Megilas Ta'anis forbids in both Adars (like Mordechai Megilah 784, Gevuras Ari here and Gra 568:7), and discusses the 13th of both Adars. Alternatively, Tosfos discusses the 13th in every year.

Support: We say so below regarding the day before Nikanor... and regarding Yom Tov Turainus. Since they were Batel from their Yom Tov (i.e. Adar 12, Yom Turainus), we do not forbid them due to the Yom Tov after them, i.e. due to being the day before [Yom Nikanor].

Similarly, since the day before Purim (Yom Nikanor) was Batel due to itself (its own Yom Tov), and permitted to fast, it is not forbidden due to the day before Purim.

Explanation: We rule that he fasts and finishes, i.e. regarding Rosh Chodesh;

It cannot refer to Chanukah and Purim. One may not fast, for it says that it is a day of Mishteh and Simchah.

Note: Megilas Ta'anis forbids fasting on both of these, but Mishteh and Simchah is written only regarding Purim, in Megilas Esther. Darchei Moshe (OC 670) brings from Mordechai ha'Aruch that Yemei Chanukah were fixed to be days of Mishteh due to Chanukas ha'Mizbe'ach (Cheshek Shlomo. See also Mitzpas Eisan.)

Explanation: This is even on the first fasts, for the severity of the last fasts applies to them.

Question: "We are Masri'a" connotes that we are Masri'a and fast. Why did the Mishnah below need to specify that that city fasts and is Masri'a, since Stam Masri'a connotes both of them?

Answer: It is because R. Akiva needs to argue in the Seifa, and teach that they are Masri'a, but do not fast.