The Gemara concludes: And the halakha is not in accordance with any of these halakhot; rather, it is in accordance with that which Rabbi Yehoshua ben Levi said: On Yom Kippur that falls on Shabbat, one who recites the day’s closing prayer [ne’ila] must mention Shabbat even in that prayer, although ne’ila is not recited every Shabbat. The reason for this is that on Yom Kippur, the day itself is obligated in four prayers, i.e., morning, additional, afternoon, and closing. When it occurs on Shabbat, one must mention Shabbat in each of the prayers. Apparently, on a day that has a unique character, that character is manifest in all sacred aspects of the day; those engendered by the day itself as well as those engendered by other factors.

The Gemara challenges this: It is difficult, as there is a contradiction between one halakha and another halakha. On the one hand, you said that the halakha is in accordance with the opinion of Rabbi Yehoshua ben Levi. And, on the other hand, we hold that the halakha is in accordance with the opinion of Rava, which contradicts the first halakha, as Rava said: On a Festival that occurs on Shabbat, the prayer leader who descends before the ark to recite the prayer abridged from the seven blessings of the Shabbat evening Amida prayer need not mention the Festival, as, if it were not also Shabbat, the prayer leader would not descend before the ark to recite this prayer during the evening prayer on a Festival. The Gemara reverts to the previous assumption that an element that does not arise from the essential halakhot of the day is considered foreign to it and is not mentioned.

This challenge is rejected: How can you compare? There, actually, even on Shabbat, the prayer leader need not repeat the prayer, just as the prayer is not repeated any other evening. It was the Sages who instituted repetition of the prayer due to concern for potential danger. The Sages sought to slightly delay those leaving the synagogue to enable people who came late to leave together with the rest of the worshippers. This was necessary because synagogues were often located beyond the city limits, and it was dangerous to walk alone at night. This repetition of the prayer does not stem from the obligation of the day but was instituted for another purpose. However, here, on Yom Kippur, it is the day that is obligated in four prayers, and therefore on each day that there are added prayers, one must mention the events that occurred on that day in those prayers just as he does in all the standard prayers.

And we learned in the mishna that one may not light with the sheep’s tail or with fat. Naḥum the Mede says that one may light using cooked fat. And the Rabbis say that one may not light with it whether or not it is cooked. The Gemara asks: Isn’t the opinion of the Rabbis identical to the unattributed opinion of the first tanna in the mishna? The Gemara answers: The practical difference between them is with regard to what Rav Beruna said that Rav said that one may light with cooked fat to which oil was added. One of the tanna’im accepts this opinion as halakha and permits lighting with it, and the other prohibits it, and the opinions are not defined. Although it seems from the formulation of the mishna that they differ on this point, it is unclear what the opinion of each tanna is.

MISHNA: In continuation of the previous mishna, this mishna adds that one may not light with burnt oil on a Festival, as the Gemara will explain below. With regard to lighting Shabbat lamps, there were Sages who prohibited the use of specific oils. Rabbi Yishmael says that one may not light with tar [itran] in deference to Shabbat because tar smells bad and disturbs those in the house. And the Rabbis permit lighting with all oils for lamps as long as they burn properly; with sesame oil, with nut oil, with turnip oil, with fish oil, with gourd oil, with tar, and even with naphtha [neft]. Rabbi Tarfon says: One may light only with olive oil in deference to Shabbat, as it is the choicest and most pleasant of the oils.

GEMARA: With regard to the statement of the mishna that one may not light with burnt oil on a Festival, the Gemara asks: What is the reason for this? The Gemara answers: Because, in general, one may not burn consecrated items on a Festival. With regard to the fundamental principle that one may not burn consecrated items on a Festival, the Gemara asks: From where are these matters derived? Ḥizkiya said, and one of the Sages from the school of Ḥizkiya taught the same, that which the verse said: “And you shall let nothing of it remain until morning; but that which remains of it until morning you shall burn with fire” (Exodus 12:10), requires explanation. As the Torah did not need to state until morning the second time. It would have been sufficient to state: But that which remains of it you shall burn with fire. Rather, why does the Torah state until morning? The verse comes to provide him with the second morning for burning. Leftover meat of the Paschal lamb is not burned on the following morning, which is a Festival, but rather on the following day, the first of the intermediate days of the Festival. From there it is derived that burning consecrated items on a Festival is prohibited.

Abaye said: This is derived from another verse, as the verse said: “This is the burnt-offering of each Shabbat on its Shabbat” (Numbers 28:10). Only the burnt-offering of Shabbat is sacrificed on Shabbat, and not a weekday burnt-offering on Shabbat, and not a weekday burnt-offering on a Festival. Apparently, performing this mitzva is prohibited even on a Festival, since it was not explicitly enumerated among the actions permitted on a Festival.

Rava said: This is derived from a different verse, as the verse said with regard to the laws of a Festival: “No manner of work shall be done in them, save that which every man must eat, that alone may be done by you” (Exodus 12:16). From the word that, it is derived that for sustenance, one is permitted to perform prohibited labor on a Festival, but not for facilitators of sustenance. Although cooking is permitted, actions that involve prohibited labors for the purpose of facilitating cooking are prohibited. From the word alone, it is derived: And not circumcision performed not at its appointed time, i.e., a circumcision may be performed on a Festival only if it is on the eighth day. A circumcision that was postponed may not be performed on a Festival. It is possible that license to perform the postponed circumcision on a Festival could have been derived by means of an a fortiori inference. Therefore, the verse explicitly prohibited doing so. The same is true with regard to burning consecrated items. Although the Torah commands burning consecrated items, it was not permitted on a Festival since there is no obligation to do so specifically on that day.

Rav Ashi said: It is derived from a different source. In the verses that speak of the Festivals, as opposed to the term Shabbat, the term shabbaton (Leviticus 23:24) appears.