Note that it does not mean that \alpha+t_{\nu,\alpha'}=\alpha'. For example, the region \alpha defined by the set of constraint \{0< x< 1, 0< y<1, x< y\} has the time-successor \alpha' defined by the set of constraint \{0< x< 1, y=1\}. Indeed, for each \nu\in\alpha, it suffices to take t_{\nu,\alpha'}=1-\nu(y). However, there exists no real t such that \alpha+t=\alpha' or even such that \alpha+t\subseteq\alpha'; indeed, \alpha defines a triangle while \alpha' defines a segment.
