  A generalization of forming derived and residual designs from $t$-designs to
subspace designs is proposed. A $q$-analog of a theorem by Van Trung, van
Leijenhorst and Driessen is proven, stating that if for some (not necessarily
realizable) parameter set the derived and residual parameter set are
realizable, the same is true for the reduced parameter set.
  As a result, we get the existence of several previously unknown subspace
designs. Some consequences are derived for the existence of large sets of
subspace designs. Furthermore, it is shown that there is no $q$-analog of the
large Witt design.
