H55F reverse switch wiring

[Mightymes]

Hey guys,
I'm looking for a little guidance on wiring up a reverse light for a h55f tranny. I'm working with a new EZ wire harness and a new light from a 72 FJ40. There are two wires coming from the tranny switch that have bullet connectors on them. I'm wondering if I can splice from the wiper power wire to the switch? I'm not sure where to go after that. The light has that one red wire and that's it. What is the other wire on the tranny for? Is that ground? Also, does the light ground itself when it's mounted?
thanx
Craig

 

[3_puppies]

the switch on the transmission is a push button type switch, power on 1 wire, when switch is depressed when in Reverse power comes out the other wire, then to the light to turn it on.
most likely the light is grounded thru its mounting bolt.