The way to prove it is to consider the forces on the slider when in position x. By Hooke's law, the mainspring supplies a force k0(x-x0), where k0 is the characteristic spring constant of the mainspring, and x0 is the rest position if there were no rebound spring. The rebound spring supplies a force -k1(x-x1), where k1 is the spring constant of the rebound spring, x1 is the rest position if there were no mainspring, and the minus sign is because (over the whole travel where both springs are acting) the springs push in opposite directions.

So the total spring force is k0(x-x0) - k1(x-x1).

This can be rearranged to (k0-k1)(x - (k0x0 - k1x1)/(k0-k1))

Which is the same force as would be applied by a weaker spring (strength k0-k1) with a different offset (note that the term on the right is a constant, independent of x).

Peter