Park and turn off the Craftsman riding mower. Remove the keys from the ignition.
Pop up and open the Sears Craftsman LT 1000's front engine hood. Find the riding mower's flywheel. This large circular iron part is visible at the top of the engine assembly. There is an iron cover over the flywheel held down by bolts. Remove each bolt with a socket wrench.
Pry off the flywheel cover once bolt removal is complete. Lay the cover aside.
Remove the single bolt at the flywheel's center, using a socket wrench. This bolt isn't visible until the flywheel cover is removed.
Position a flywheel pulley over the exposed flywheel so that the pulley's arms reach over the outer fins of the flywheel.
Tighten the center bolt on the flywheel pulley, using a socket wrench, until the pressure created pops the flywheel free from the engine assembly. Set the flywheel and pulley aside. This will reveal the top of the cable clutch shaft.
Find the mower's oil pan. This pan is at the bottom of the engine assembly, below where the flywheel was removed. Remove the bolts holding it in place using a socket wrench. Set the pan aside. This will reveal the bottom of the clutch shaft.
Look along the clutch shaft for the now visible bolts. Remove these bolts with a socket wrench.
Strike the top of the exposed clutch shaft with a rubber mallet. This will pop the shaft free.
Slide the replacement cable clutch shaft into the place vacated by the old one, and secure it with the previously removed shaft bolts. Tighten these into place with a socket wrench.
Follow the oil pan and flywheel disassembly procedures in reverse to finish the repairs.