The lower two lines in the lower radiator are transmission cooler lines. The transmission cooler doesn't go through the tube/fin part of the radiator, it just exchanges heat with the coolant in the radiator itself. This keeps it within normal operating temperatures - not too hot or cold. Also if you run an auxilary cooler, you still should run it through the radiator cooler in order to allow the transmission fluid to get up to normal operating temps.
All modern automatic cars have this cooler, if you ever had to buy an aftermarket radiator and you drive a car with auto or manual, you have to specify which one you have. If you need to replace this hose, it's easy. Just 5/16" transmission cooler hose at any auto parts store. Make sure you get 5/16" and not 3/8". I picked some up recently since I'm about to install an auxilary cooler and it was like $1.99 per foot.