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This is the second part of the Run For Your Life Mission in Hitman Absolution. In this part you must get through the library that is heavily patrolled by police officers to the exit. This guide explains how to do that in the hardest, purist, difficulty. It does not explain how to do all the challenges in this area.
When you first start you stand on a ledge. Drop down and go the the right and take cover behind the pillar. A police officer will walk up to the level.
After he is up there with you sneak past him down the stairs until you can take cover behind some book shelves at the bottom of the stairs.
Eventually 2 of the 3 police officers left will leave. Once they start to leave you can leave the cover and sneak past all of them behind the book shelves the remaining police officer leans on. Follow the little artificial corridor until you see some stairs. Crouch walk up the stairs and through the door to the ledge. Follow the railing all the way to the right and just keep in cover.
Wait for the police officer to have walked away from you a little bit, then go up to the fallen over book shelves. Wait for the light to go off and the two police officers to have stopped talking. Then let the police officer right in front of you walk away a little bit, then sneak past him. Once you are on the opposite side there is an empty cup board. Enter that cupboard.
Remain in it until a police officer has walked past you. If you like you can peak out. Once the police officer has walked past your hiding place wait for 4 or 5 more seconds (otherwise the police officer you snuck by before will notice you). Leave the cup board and follow the walkway around the corner and down the stairs (don't worry if you hear the being detected noise, you are fast enough that it doesn't matter). Once you are at the bottom go towards the hole in the wall opposite the stairs and climb through it.
Quickly make your way to the entry between the two book shelves. If you aren't fast enough a police officer walks into that same area and you have to wait until he leaves again.
Go up the stairs. At the top of the stairs go to the left and walk through the gap between the railing and the wall. Once you get up to the door you can just sneak past the guard standing in front of it.
Follow the stairs up and the little platform around until you can jump onto a ledge.
If you keep going forward there is a metal beam across the platform to the other side.
Go over it. Once you are on the other side turn left and you will see a door with an exit sign above it. That is the end of this part of the Run For Your Life Mission.