This is the 4998th most frequent Russian word.
лестнице
"Лестнице" (dative or prepositional case of "лестница") means "staircase" or "stairs" in English.
Here, 'лестнице' is in the prepositional case and indicates the location or instrument by which the subject is climbing.
Он поднимается по лестнице.
He is climbing the staircase.
In this sentence, 'лестнице' is in the prepositional case and refers to the location where the book is lying.
На лестнице лежит книга.
A book is lying on the staircase.
Again, 'лестнице' is in the prepositional case and denotes the staircase associated with the railing being repaired.