Higher order reduction theorems for general linear connections

The reduction theorems for general linear and classical connections are generalized for operators with values in higher order gauge-natural bundles. We prove that natural operators depending on the $s_1$-jets of classical connections, on the $s_2$-jets of general linear connections and on the $r$-jets of tensor fields with values in gauge-natural bundles of order $k\ge 1$,$s_1+2\ge s_2$,$s_1,s_2\ge r-1\ge k-2$,canbefactorizedthroughthe$(k-2)$-jetsofbothconnections,the$(k-1)$-jets of the tensor fields and sufficiently high covariant differentials of the curvature tensors and the tensor fields.