The entire time William stayed inside the library to learn everything about the Progenitor Mages, the kids of the remaining opposing party had left to follow the mysterious man who promised them strength.
Liam and the others learned about this and decided not to pursue the matter since it will only worsen the relationship between the two parties. Though it can be seen as an arrogant act by their leader, in reality it was completely different.
Because of William's dedication in learning about the truth, Liam made sure not to disturb him until he got out of the library after reading everything he needed.
[A few days have passed]
The door of the library finally opens and William steps out stretching his sore muscles after not being able to move around for a few days.