The library’s proposition is unrelated to the school budget. New York State allows association libraries to ask residents of a school district or municipality for funding by ballot proposition, which is approved by majority vote. This method gives residents a direct say in the level of funding their local library receives.
The library budget is completely separate from the school budget. The district merely collects funds approved by the voters and disburses them to the library. School taxes fund the schools, and library taxes fund the library. School and library taxes appear on the same bill and are paid at the same time, but the school district and the library are two separate entities.