When you decide to clean out underneath the old sink cabinet you may want to consider painting the area. A word of caution though, if the old sink cabinet bottom has absorbed some sort of liquid and has swelled, you may be in for a challenging task. The reason I say this is because if the substance was lotion that soaked into the flake board, you will have a very tough time getting the paint to adhere.
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An upper (wall) cupboard needs to be fastened to the wall before a decorative mirror can be added. Rather than learn how to install the mirror, I have always hired a glass company to measure, supply, deliver and attach the mirror to the back How to Install Mirror in Cabinets