Go to the middle door on the lower level, north side (marked B in the diagram in the main walkthrough) and press Action to push it open.

Follow the passageway to a climbing pole. Unlike the other poles you've seen so far, this one has a gear-like blade moving around the square opening. To get through without taking damage, stand at the edge of the opening, to either side of the pole and hop back once.

Now watch the blade. If Lara is standing on the left side of the hole, wait for the blade to move to the right (or vice versa).

Then hold Action and either take a standing jump forward or just run off the edge...

...to land safely in the room below.

Shoot the large vase to get a small medipack.

Then repeat the hole maneuver to descend past the next blade to the level below.

Shoot the vases here to get shotgun shells...

...and a large medipack. Then repeat the maneuver a third time or just safety drop carefully past the next blade into the passageway below.

Follow the hallway to a second pole. There's no need to climb it.

Just stand below one corner of the opening and wait for the blade to pass to the other side.

Then pull up through the opening.

[Lost Library Walkthrough]