Decomposition of a jet bundle of differentiable mappings into a Whitney sum of tangent bundles

We prove that the bundle $J_m^rM_n$ of jets of differential mappings of open neighbourhoods of $0\in\mathbb R^m$ into differential manifold $M_n$ can be decomposed into the Whitnev sum $\bigoplus\limits_{a=1}^NT_a(M_n)$, where $N=\binom{m+r}r-1$. To get such a decomposition $J_m^rM_n$ it is sufficient to take a linear connection on $M$. We use this decomposition to construct lifts of linear forms, vector fields and Riemannian metrics from the base $M_n$ into the bundle $J_m^rM_n$.