Welcome to OSM!

The relation of the Barr Smith reading room is wrong; multipolygon outer ways cannot share segments... I take it the two member ways have different levels??? Tag each way as builing part with the correct levels .. and then...

The Way: Barr Smith Library (678979212) looks to be the 'reading room' ... and would be best tagged with amenity=library.. and 'reading room'??? 

