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One removes two Sifrei Torah from the ark. Five people are called to 1st Torah and the 6th to Maftir. [On Shabbos one reads 7 Aliyos from the first Torah.]
Moshe Rabbeinu established for the Jewish people that on every Yom Tov one is to read the Torah portion dealing with that Holiday. The Sages of the Mishneh and Gemara chose the exact portion of the Torah that is read on each holiday.
Haftorah: The reading of the Haftorah was instituted by the Anshei Kneses Hagedola. They instituted that one should read from Navi on every holiday from a portion that deals with the holiday events. The Sages of the Mishneh and Gemara chose the exact portion of Navi that is read on each holiday.
The Maftir: [Maftir in truth refers to the reading of the portion from Navi formally known as the Haftorah. It comes from the word Petor which means finish, as the reading is completed with the Haftorah.] However, the person that reads the Maftir from Navi is also required to first read a section from the Torah. In the times of the Sages of the Mishneh and Talmud the Maftir would read from the Torah as part of the five Aliyos that were read from the portion of that Holiday which was read from the first Torah scroll. [There was hence no second scroll taken out from the ark] However, in later generations, the Rabanan Savuraiy established, and so was accustomed by the Geonim which followed them, that the Maftir is to read [from a second scroll] the portion of the Musaf sacrifice written in Parshas Pinchas. This institution was based on the following teaching of the Sages: Avraham stated in front of Hashem “Master of the world when the Temple is not in existence the children of Israel do not have on what to find support. What will be with my children when they sin? Perhaps you will do to them as you did to the generation of the Mabul and Haflaga. Hashem answered “I have already established for them the order of the Karbanos. When the Karbanos are read, I consider it as if they sacrificed the Karban before me and I forgive all of their sins. [Regarding why the Maftir is read as the sixth Aliyah and is not part of the five Aliyos-see chapter 282/10-13!]
Two Torah scrolls: Being that two different sections of the Torah need to be read on Yom Tov, therefore, one is initially required to remove two Torah scrolls; one for the Holiday reading, and the second for the Maftir. It does not suffice to remove only one Torah scroll and then roll it to the Parsha of Musafim [for Maftir]. [The scrolls are to be rolled to their proper places prior to Davening. Doing so is not to be delayed until Kerias Hatorah. If only one scroll is available then the two portions are to be read from the same scroll.]
Yud Gimmel Middos-Hashem Hashem:
When the Shalosh Regalim fall on a weekday, the 13 attributes [i.e. passage of Hashem Hashem] is recited one time when the Ark is opened to remove the Torah scrolls, prior to Berich Shmei. If Yom Tov falls on Shabbos, the Yud Gimmel Middos [and Ribono Shel Olam] is not recited. After the recital of Hashem Hashem one recites the prayer of Ribono Shel Olam [one time]. [Likewise, the verse of Veani Sefilasi, which is recited after the Ribono Shel Olam is only recited one time.] This is then followed by the prayer of Berich Shmei.
 Admur ibid; 282/1; M”A 135/1; 535; Rambam Tefila 13/8; Mishneh Megillah 31a
The hint in scripture: The above institution is hinted to in the verse “Vayidaber Moshe Es Moadeiy Hashem El Bnei Yisrael”, from which we learn that Moshe established that each Parsha of the Moadim is to be read at its proper time. [Admur ibid]
 Admur ibid; Megillah ibid
 Admur 282/1; 488/5; M”A 135/1
 The reason: The reason behind the different number of Aliyos [Three during the week; Four on Rosh Chodesh; Five on Yom Tov; six on Rosh Hashanah; seven on Shabbos] is based on the levels of holiness that each day contains over the other days. See 282/1 for further details on this matter.
 Admur 488/5
 Admur ibid; Sefer Hamachria 31 in name of Rabbeinu Tam that it was instituted by Ezra
 Admur 488/6
 Admur 282/10; 488/6
 Admur 488/6; Michaber 144/3
 One must read from the Parsha of the Holiday, as per the institution of Moshe Rabbeinu, and one must also read the Karbanos from Parshas Pinchas, as per the institution of the Rabanan Savuraiy.
 The reason: As one may not roll a Sefer Torah in front of the congregation due to respect of the congregation. [Admur ibid]
 See M”B 144/13
 Michaber 144/3; Although the scroll will need to be rolled to the proper place of the second reading, after completing the first reading, nevertheless since there is no other scroll available we allow it to be rolled and differ the “respect of the Tzibur”. [ibid; See M”B 144/16]
 Admur 490:11; 282:11
 The reason: This is done to differentiate between the original institution to call up 5 people on Yom Tov, and the additional institution to call up an additional person for Maftir. [Admur ibid]
 Siddur Admur
 Admur ibid; Peri Eitz Chaim Shaar 20/3
It is not customary to recite the words “Vayavor Hashem Al Panav” prior to saying it being that the Hashem Hashem is recited as a prayer and not as a verse in the Torah. [Halichos Olam brought in Piskeiy Teshuvos 584 footnote 16, unlike Mateh Ephraim 584/584/16; Otzer Minhagei Chabad 206].
 Darkei Chaim Veshalom 715; Siddur Rav Raskin footnote 72
 Implication of Admur in Siddur; Shaarei Ephraim 10/5 See Sefer Haminhagim p. 119 [English Edition]; Hayom Yom 10th Tishrei; Hisvaadus 6th Tishrei 1970; Shaarei Teshuvah 488/1; Likkutei Maharich Pesach based on Ramaz; Custom brought in M”E 584/16; Luach Eretz Yisrael; See Otzer Minhagei Chabad 210; Piskeiy Teshuvos 488/3
 This prayer was established with Ruach Hakodesh by the Arizal. [Piskeiy Teshuvos 584/3; See Kaf Hachaim 134/11; Shaar Hakavanos p. 89] On Rosh Hashanah the words “Umechol Al Kol Avonosaiy” is omitted as we do not mention sin on Rosh Hashanah. [See Siddur Admur; Otzer Minhagei Chabad 205]
 Shaarei Teshuvah 488 [“so is the custom”]; implication of Admur in Siddur [Rav Raskin p. 300]
Other Opinions: According to the Arizal one is to recite the prayer of Ribono Shel Olam three times. [Peri Eitz Chaim Shaar Tefilos R”H]
 Implication of Admur in Siddur and so is the Chabad custom
Other customs: Some have the custom to recite the verse of Veani Sefilasi three times. [Peri Eitz Chaim Shaar R”H; Siddur Shlah]
 Siddur Admur; Ketzei Hamateh 619/48 [so is custom of world] unlike the custom brought in Shaarei Ephraim 10/5 and Mateh Ephraim 619/48