We work in the Alternative Set Theory. Let us fix some set-defined sequence. Two cuts of N we call terminal equivalent, if our sequence has the same limits with a respect to both of them. The aim of the work is to get conditions on a cut of N, under which this cut permits nontrivial #shifting#, preserving terminal equivalence, i.e. we can slightly enlarge or slightly diminish it with all proper cuts between two so obtained horizons be terminal equivalent.