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Let's assume S is a system (inertia) and S' is relatively moving to S with velocity v. Now I can take the Lorentz-transformation to get from S into S':

x'=gamma(x-vt) and t'=gamma(t-beta*x/c)

Now we take an event in S starting at t1 and ending at t2, thus taking the time dt=t2-t1 in S. I want to know how an observer in S' thinks about the time of the evening. Ok, using the Lorentz-transformation we yield:

dt'=t2-t1=gamma(t2-beta*x/c)-gamma(t1-beta*x/c)=gamma(t2-t1)

BUT SINCE GAMMA>1 IT FOLLOWS THAT dt'>dt and thus if the event in S takes 5s the event may take (if v is sufficient large) 10s in S'. But that would mean more time in S' has passed than in S. BUT WE KNOW, THAT IS SIMPLY NOT TRUE. All books tell one should use the following transformation:

dt'=gamma^-1*dt, but WHY (I mean the Lorentz-transformation is defined in another way)???

Thanks for your answer