lim+, δ+, and Non-Permutability of β-Steps

Using a human-oriented formal example proof of the (lim +) theorem, i.e. that the sum of limits is the limit of the sum, which is of value for reference on its own, we exhibit a nonpermutability of β-steps and δ+-steps (according to Smullyan’s classification), which is not visible with non-liberalized δ-rules and not serious with further liberalized δ-rules, such as the δ+ + -rule. Besides a careful presentation of the search for a proof of (lim +) with several pedagogical intentions, the main subject is to explain why the order of β-steps plays such a practically important role in some calculi.