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(Some of this is drawn from https://www.halachipedia.com/index.php?title=Animals_on_Shabbat) I think Sefaria's language of "Sabbath-observant animal" was a bit confusing. Any animal owned by a Jew may not be prompted to do activity prohibited for a Jew themselves to do on Sabbath. Let's break that statement down, and hopefully all your questions ...


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David's seeing Batsheva bathing was accidental and unintentional - the story is detailed in Sanhedrin 107a: Bathsheba was shampooing her head behind a screen, which concealed her from sight. Satan came and appeared to David as a bird. David shot an arrow at the bird, the arrow broke the screen, Bathsheba was exposed, and David saw her.


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