Proof that summation of N Markov processes is a Markov process

Hi all,

I have this assignment where I should prove the following:

Let X1(t),...,XN(t) be N independent Markov processes with two states each {0,1}. The instantaneous transition rates are the same in all N Markov chains and equal to q01 = lambda and q10 = mu. Consider the stochastic process defined as Z(t) = X1(t) + ... + XN(t).

Show that Z(t) is a Markov process.

I know this means I should prove the following:

P(Z(t+s)=j|Z(s)=i,Z(u)=z(u),0<=u<s)=P(Z(t+s)=j|Z(s )=i)

But I have no idea where to go from here. Is it necessary to use the instantaneous transition rates for Z(t)? Which i guess would be LAMBDA(t)=NumberStatesOne(t)*lambda and MU(t)=NumberStatesZero(t)*mu.

Maybe someone could give me a push in the right direction?