  In this paper we prove optimal error estimates for {solutions with natural
regularity} of the equations describing the unsteady motion of incompressible
shear-thinning fluids. We consider a full space-time semi-implicit scheme for
the discretization. The main novelty, with respect to previous results, is that
we obtain the estimates directly without introducing intermediate semi-discrete
problems, which enables the treatment of homogeneous Dirichlet boundary
conditions.
