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1. The Gemara discusses the principle that one may assume that a Shali'ach fulfilled his assignment when it applies to a d'Oraisa law.
2. Rav Nachman and Rav Sheshes cite proofs for their opinions (#1).
3. We may assume (when in doubt) that Terumah has been separated from a person's produce when that person is known to be scrupulous about separating Terumah.
4. There is a dispute about whether it is better for a Chaver to transgress a minor matter in order to prevent an Am ha'Aretz from transgressing a severe matter.
5. The Mishnah discusses when one's Eruv is valid despite it having been put it in a tree.
A BIT MORE
1. Rav Nachman: This principle applies only to d'Rabanan laws, but not d'Oraisa laws. Rav Sheshes: This principle applies to both d'Oraisa and d'Rabanan laws.
2. For example, Rav Sheshes notes that people who are far away from Yerushalayim may eat new grain after midday on the day that the Korban ha'Omer is brought, even though they are not certain that it was brought yet. Rav Nachman refutes this proof by saying that while we may assume that Beis Din fulfilled its mission, we may not assume that a Shali'ach fulfilled his mission.
3. Rav Nachman notes this is not the related to the principle that we may assume that a Shali'ach fulfilled his mission, because here it applies only to a person who is always careful about separating Terumah. Rashi explains that the person knows that it is his responsibility to separate Terumah from his fruits, while a Shali'ach might not believe that the person who sent him really depends on him.
4. Rebbi: It is better for the Chaver to commit the small sin to save the Am ha'Aretz from transgressing a severe one. Raban Shimon ben Gamliel: One should not commit even a small sin to prevent another person from performing a severe sin (see Tosfos).
5. The Gemara quotes Rav Nachman and others who say that even though the use of a tree is forbidden on Shabbos, since Rebbi understands that the Rabanan did not apply their prohibitions during Bein ha'Shemashos (which is when the Eruv takes effect), the Eruv is effective.
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