At the Shop buy a Super Potion or two. These come in handy. Head up town and you'll see a man in a lab coat walk out his door and fall. He asks for help. Just leave him lying and enter the building he came out of. You'll hear a brief word that Silva's gone too far and the Colosseum is out of power. You have to find Silva who has the Gear.
Silva has actually already left the city and is at the Construction Lot. Head there. At the top-right you'll see Silva has already left, but he left or hid the gear here. It's at the top-right corner, you can't miss it. Grab it and head back to Pyrite. Head all the way back to the windmill building. Duking is there when you arrive. Duking is a bit torn that Silva has betrayed his trust. Duking becomes fond of you two when he finds you have the Gear. Walk up to the gears and place the missing Gear. The Windmill is working again and the Colosseum now has power.
Right before you leave Duking asks for a little help because he thinks you're no "ordinary trainer". It's time to go upstairs and to the Colosseum!
