Hullin 12: Can You Trust the Shochet?

Do you need to see the entire process of shechitah to be able to trust that it was done properly? The Gemara has a case that does require one to see the entirety of the process, but we know that that isn't usually how it works (so often, we do not see any of the process of shechitah!), so the sages delve into the particulars of the case. Perhaps it's a question of whether you know the shochet's level of knowledge and expertise. Also, if one chooses an agent to go slaughter an animal and the agent discovers that the animal was already slaughtered - can it be trusted to be a kosher shechitah? What about terumah that was set aside? Would it have been set aside properly?