The 582 was/is gear drive with a Briggs engine.
So a 782 transmission would be make more sense in order to have hydraulics for the loader.
Question is, do you have a Briggs engine, or a Kohler which would have come in a 782??
More likely, it is not the original hood.
What is the serial number on the plate down by the brake pedal??
Is the tractor red or yellow/white??
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