I'll just go purely off my circuitry knowledge here, sorry I cant seem to find too much pertinent information in research because we are lacking model numbers.
1st off, you found the safety switch under the seat right?
Safety switches for riding lawn mowers are typically nothing more than on/off plunger switches. When no one is on the seat the plunger is up and the switch is open/off. When someone is on the seat, the plunger will be pushed in and will cause the switch to become closed/on...completing the circuit. So all you have to do is connect the two wires that are plugged into the switch and you should by pass it.
NOTE: The above explanation assumes a "fail-open" switch and that you need a closed circuit to run the engine.
POST NOTE: Definitely don't recommend bypassing the switch. Safety equipment is there for a reason. Odds are you could rig up a $2 radio shack switch if you're concerned about money :)
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