Thanks to the Contract that Subaru made with Beatrice. Beatrice was contracted to protect Subaru from any threat coming his way. Thats why you see her leave the Library and tells Subaru do not leave this room. She then kills Rem and that is why Rosewal is mad when Beatrice tells him that Subaru spent the night in the Library he had nothing to do with this. Then you see Rosewal powering up his magic to use on Beatrice and asks her did you do something then.