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In this paper the authors compare two different approaches to the second order absolute differentiation of a fibered manifold (one of them was studied by the authors [Arch. Math., Brno 33, 23-35 (1997; Zbl 0910.53014)]. The main goal is the extension of one approach to connections on functional bundles of all smooth maps between the fibers of two fibered manifolds over the same base (we refer to the book ``Natural Operations in Differential Geometry'' [Springer, Berlin (1993; Zbl 0782.53013)] {\it I. Kolar, P. Michor} and {\it J. Slovak}, as an introduction for some previous results used in this article).
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