Explanation #1 (Rashi in Chulin 13a): E.g. he was engaged with a knife to lift it or throw it. (He did not intend for Shechitah.)

Objection: In such a case even [Shechitah of] Chulin is Pasul according to Rabanan who argue with R. Nasan and require intent!

And even intent to cut, e.g. he threw a knife to insert it in a wall, which is considered intent for cutting, like is proven in Chulin (31b), even so Rabanan disqualify;

[They disqualify] unless he intends to cut Simanim. [Then,] even though he does not intend for Shechitah to permit the meat, surely they are Machshir, like it says there "Rabanan, granted they do not require intent for Shechitah, they require [intent] to cut;

Explanation: They require intent to cut Simanim, but another cutting is not [enough], for throwing a knife to insert it in a wall is considered intent for cutting, and Rabanan disqualify.

Explanation #2: Mis'asek in Kodshim is that he intends to cut Simanim, but not for the sake of Shechitah. For Chulin it is Kosher, but for Kodshim it is Pasul.

Explanation #3: There is another Mis'asek, e.g. for the sake of Chulin, that he thought that the animal is Chulin. We learn from all of these from "v'Shachat Es Ben ha'Bakar."

Inference (R. Efrayim): A knife is not required for Shechitah, because the Mishnah did not teach that "they must be slaughtered with a Kli Shares in the north", like it taught about Kabalah "Kabalah of their blood is in a Kli Shares in the north."

Support #1: According to this, it is fine what it says in Chulin (3a) "e.g. he checked a sharp reed and slaughtered with it", for a Kli Shares is not required.

Support #2: It connotes a little like this in Pesachim (66a), that everyone used to bring his knife [for Shechitas Pesach] from his house.

Rejection #1: This cannot be! In Menachos (82b) and below (97b) we expound from "va'Yikach Es ha'Ma'acheles" that Olah requires a Kli, and we learn all [Zevachim] from it.

Rejection #2: In Sotah (14b) it connotes that the knife is Mekadesh the blood.

Rejection #3: In Menachos (78b) there is an opinion that a [Shechitah] knife is greater than a [Stam] Kli Shares. Even though [a knife] has no interior, it is Mekadesh.

Implied question: We find (Chulin 3a) that Shechitah with a sharp reed is Kosher!

Answer: We can say that he fixed it and made it a Kli, and according to R. Yosi b'Ribi Yehudah, who says in Sotah (14b) that we can make a Kli Shares from wood [or other vegetation].

Implied question: In Pesachim (66a it connotes that everyone used to bring his knife for Shechitas Pesach from his house)!

Answer: We can say that everyone was Makdish his knife before Shabbos. (There it discusses when Erev Pesach was on Shabbos.)

Implied question: Why didn't the Mishnah teach "they are slaughtered with a Kli Shares in the north", like it taught about Kabalah?

Answer: These are different. Also regarding Kabalah, it mentioned the Kli only because also the Kli must be in the north. This excludes if the animal is in the north and the Kli is in the south and the blood spurts into it.

Proof: In all of these (i.e. Kodshei Kodoshim) that require the north, it teaches Shechitah... and Kabalas Dam in a Kli Shares in the north, but regarding Kodshim Kalim, which are slaughtered anywhere in the Azarah, it does not teach Kabalas Dam in a Kli Shares, even though Kabalah require a Kli Shares!

Support: Also above (26a), a Beraisa teaches that Kodshim Kalim are slaughtered inside [the Azarah], and Kabalas Dam is in a Kli Shares inside, it comes to teach that the Kli must be inside;

This excludes if most of the Kli is outside and the minority is inside, even though the blood inside is not disqualified though Yotzei, for it is received in the minority [of the Kli] inside.

Alternatively, it says so because there is Asham Metzora, in which [some] of the blood is received in the hand. Therefore, it needs to teach in all Kodshei [Kodoshim] that Kabalas Dam is in a Kli Shares.

Question: The Tosefta in Menachos teaches that there is a stringency of Kemitzah over Shechitah. Kemitzah requires a Kli, and Shechitah does not require a Kli, rather, even a sharp reed!

You are forced to say that it does not mean that it does not require a Kli at all, for we say "just like Olah requires a Kli...!" Rather, it means that it does not require a Kli Shares.

Answer: We can explain that it means that we are not concerned for a nice Kli, and we do not disqualify a sharp reed due to Hakrivehu Na l'Fechasecha (one would not give an inferior gift to a governor to find favor in his eyes), like the Gemara asks in Sotah (14b) about a Kefifah Mitzris (wicker basket, in which Minchas Sotah is brought);

Explanation #1: This is because it is a gift. It is normal to bring it in a nice Kli. However, we are not concerned about [the Kli for] Shechitah.

Explanation #2: Alternatively, a Kefifah Mitzris is more repulsive than a sharp reed.

Explanation #1 (Rashi): We learn Par Yom Kipur from a Hekesh of Par Helam Davar, like it says above (39a) "l'Par" - this is Par Yom Kipur.

Question: Something learned from a Hekesh does not return to teach through a Hekesh! It does not say "El Beis ha'Deshen" regarding Par Helam Davar itself. It is learned from Par Kohen Mashi'ach;

It says about Par Helam "Ka'asher Saraf Es ha'Par ha'Rishon." This is a Hekesh, like "v'Shachat ha'Chatas bi'Mkom ha'Olah." This is called a Hekesh below (48a).

According to Rashi, who explained above (40a) that we learn from Par Yom Kipur from Par Kohen Mashi'ach regarding "Es", "b'Dam" and Tevilah, if so, we can say that we can learn also burning in Beis ha'Deshen. However, according to R. Tam there [who says that all are equated to Par Helam, and Par Helam Davar is equated to Par Kohen Mashi'ach], it is difficult!

Answer: There are two Hekeshim of Par Helam - "v'Asah Ka'asher Asah l'Par", which we expounded above (40a) to refer to Par Kohen Mashi'ach, to make the Haza'os Me'akev in the Haza'os of Mashi'ach, and also to teach about Yoseres ha'Kaved and the two kidneys in Par Helam from Mashi'ach...

Likewise, we can learn also burning, Par Helam from Mashi'ach. When the Torah returned to write about Par Helam "v'Cha'asher Saraf", it is as if it wrote burning in it itself.

Explanation #2: Burning outside of three Machanos, we learn below (105b) from the three times it is written "mi'Chutz la'Machaneh" to teach a third Machaneh, as if it was written in each;

This is like a mere Giluy Milsa that "mi'Chutz la'Machaneh" written in each one refers to a third Machaneh;

Therefore, we properly learn Par Yom Kipur from Par Helam, and mi'Chutz la'Machaneh" written there regarding Par Yom Kipur is extra according to Rabanan, to teach that once it left one Machaneh, one who burns it is Metamei his garments.

And according to R. Shimon [it is extra] for a Gezeirah Shavah to teach from Parah. Just like here [it is burned outside of] three Machanos, also there three Machanos;

Just like there it was east of Yerushalayim, also here it was east of Yerushalayim.

Question: However, Beis ha'Deshen is surely difficult. Granted, it is as if three Machanos were written [about Par Helam] itself. Beis ha'Deshen is not written about it itself. It is possible that [Par Helam] is burned outside of three Machanos, not in Beis ha'Deshen!

If so, the requirement of the place of the ashes is something learned from a Hekesh. It cannot return to teach through a Hekesh!

Answer #1: Perhaps also this is considered a mere Giluy Milsa, that everything burned outside of three Machanos is burned in one place.

Support: Now it is fine [how we learn about] Se'ir Yom Kipur, which is not equated to Par Helam Davar.

Answer #2: For all of them, we learn Beis ha'Deshen through a Gezeirah Shavah "mi'Chutz la'Machaneh" from Kohen Mashi'ach.

Implied suggestion: Perhaps we learn from the inclusion "Chatas... Chatas" written regarding Par and Se'ir Yom Kipur;

We expound below (83a) "v'Es Par ha'Chatas v'Es Se'ir ha'Chatas Yotzi El mi'Chutz la'Machaneh v'Sarfu." Why does it say Chatas... Chatas? One might have thought that only these are Metamei Begadim [of those who burns them] and are burned in Beis ha'Deshen;

What is the source to include Par Kohen Mashi'ach, Par Helam Davar of the Tzibur and Se'irei Avodah Zarah? It says "Chatas... Chatas." R. Yehudah says so.

Rejection: You are forced to say that "they are burned in Beis ha'Deshen" was taught needlessly. "Shefech ha'Deshen" is written primarily regarding Par Kohen Mashi'ach!

Rather, it was taught due to Metamei Begadim, which is not written regarding these.

Proof: We must say so, for R. Meir argues with [R. Yehudah], and says "this is not needed. It says "Lechaper". Why does it say Lechaper? This teaches that everything that is Mechaper inside [the Heichal], one who burns it is Metamei Begadim.

Defense (of Suggestion): Perhaps just like he learns from the inclusion Chatas... Chatas Tum'as Begadim for all of them (Parim and Se'irim ha'Nisrafim) from Par and Se'ir Yom Kipur, likewise he learns from them, through that inclusion, Beis ha'Deshen for Par and Se'ir Yom Kipur.

Explanation #1 (Rashi): This is because it says "Etzba" regarding oil, and we learn blood from it.

Objection: If so, this is a Hekesh from a Binyan Av. Below (51a), we ask [and did not resolve] whether something learned from a Binyan Av returns to teach through a Hekesh!

Explanation #2: Also Lekichah from Nesinah is a Binyan Av, and we learn from its Nesinah, and not from other Korbanos;

Below, we say that we properly learn a Binyan Av from a Binyan Av. This is from R. Baruch.

Question: Also regarding Chatas, it is written "v'Lakach... v'Nasan", and we do not say that just like Nesinah is with the Kohen's body, also Lekichah is with the Kohen's body!

Answer #1: Perhaps here he expounds because it is extra. It is written about Metzora Ashir and Oni.

Implied question: In Toras Kohanim, it expounds from the first (written about Metzora Ashir. The latter, written about Oni, is extra!)

Answer: It relies on the latter (really, we expound the latter, which is extra).

Answer #2: It seems to me that since regarding Chatas it is written "b'Etzba'o v'Nasan", he does not receive with his hand, for it is not possible to receive blood on his finger!

Rather, you are forced to say that it means [that he receives] in a Kli. "B'Etzba'o" shows that we make a handle on the brim of the bucket, like it says above (24a);

However, regarding Asham it is not written "b'Etzba'o", lest we say that v'Lakach b'Etzba'o is dipping [his finger in the blood in order to put it on the Metzora], and it does not discuss Kabalah;

Therefore we need the Gezeirah Shavah. We learn from "va'Yikach [Moshe] Chatzi ha'Dam va'Yasem ba'Aganos", in Toras Kohanim. This is from R. Baruch.

Explanation #1 (Rashi): We learn [from a Gezeirah Shavah] Lekichah-Lekichah in Toras Kohanim. It is written about Chatas "v'Lakach ha'Kohen mi'Dam ha'Chatas", and it is written about Olas Sinai "va'Yikach Moshe Chatzi ha'Dam va'Yasem ba'Aganos."

Objection: This is something learned from a Gezeirah Shavah, and it returns to teach through a Hekesh "ka'Chatas ha'Asham." This was a question below (50a), and it was not resolved according to the opinion that it depends on the source (whether or not it is Kodesh)!

Explanation #2: There was no need for Rashi to explain that we learn from Lekichah-Lekichah. Below (97b), it connotes that we learn from a Hekesh Zos ha'Torah from Zivchei Shalmei Tzibur;

Citation (97b): Just like Olah requires a Kli, also every [Zevach] requires a Kli. What Kli does this discuss? If it is a bucket, also regarding Zivchei Shalmei Tzibur it says "va'Yikach Moshe Chatzi ha'Dam va'Yasem ba'Aganos"!

Explanation: Since [the Torah] returned to equate all of them to Shelamim, which is written in this verse, and we learn from it there "just like Shelamim is Mefagel and becomes Pigul..." also a Kli we should learn from Shelamim, for also there at Sinai ("va'Yikach Moshe...") was Shelamim!

Question: In any case it is still difficult, for this is something learned from a Hekesh, and it does not return to teach through a Hekesh!

Answer: The Hekesh Zos ha'Torah, also Asham is written in it. It is equated also to Shelamim, and all Ashamos are included, even Asham Metzora;

Rather, one might have thought that the Torah excluded [Asham Metzora] from Kabalah in a Kli, and we expound "b'Yad" that [Kabalah] is through the Kohen's body itself. Ka'Chatas ha'Asham reveals that the verse does not come to totally exclude Kabalah in a Kli. (Rather, some Kabalah is in his hand, and some is in a Kli.)

Question: Why do we have the Gezeirah Shavah of Lekichah from [Korbanos of] Sinai? We should learn all of them from Shelamim!

Answer: We learn Chatas from Shelamim for Kabalah in a Kli from a Hekesh, and the Gezeirah Shavah Lekichah-Lekichah is needed so it is as if it was written in Chatas itself, and then we learn Asham from Chatas through the Hekesh ka'Chatas ha'Asham.

Implied question: Why do we not learn initially Asham from Shelamim through the Hekesh Zos ha'Torah, like we learn regarding Pigul?

Answer: If we did, what would we learn from the Gezeirah Shavah Lekichah-Lekichah?!