I'll make the assumption that the engine is killed by grounding the black wire. This is typical in small engines, i.e. grounding the magneto to kill ignition. So: Kill=Black wire grounded, Run=Black wire open
The Green wire is the primary ground for the starter solenoid. Also, since the seat switch is normally Open, we can deduce that the Brown and Green wires must be grounded to for the mower to start and run.
All you need then, is a circuit that keeps the Black wire open (not grounded) as long as both the Brown wire and Green wire are grounded. If either the Brown or Green wire is not grounded, the Black wire should be grounded.
I'd do this with relays, as in this circuit I'll try to attach. Choose R1 and R2 according to the requirements of the relay coil. They could also be the same resistor. The relays keep the Black wire grounded (kill) unless both relays are energized by having their coils grounded by the Brown and Green wires. You could do the same thing with semiconductors but that is a horrendous electrical enviornment....keep it simple.
Be sure to verify my assumption that Kill=Black wire grounded.