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Turn the barstool upside down, and use a flat-head screwdriver to remove the staples holding the fabric in place.
Pull the fabric off of the barstool, and use it as a template for cutting a new piece of upholstery fabric.
Lay the fabric face down, and place the stool upside down on top of it. Center the stool on the fabric, and begin rolling the fabric over the edges of the seat. Staple the fabric in place.
For corners, pull the corner material away from the seat. Press the sides in toward each other, and fold the material over the edge of the seat. Cut off any excess fabric, and staple the corners.