After you reach the Facilities' mainframe there will be a boss-battle with a group of nightkin. Nightkins are mutants that can cloak their bodies so you can't see them. A laser-based weapon works best, because of its ability to shoot more than one enemy at a time. There should be about eight to ten of them. After you've gotten rid of them, access the computer in the center of the room. With the computer you can open the vault's ventilation shafts. Do that and head out the western door back to the garden.
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