Differentiation on Interval

Summary This article generalizes the differential method on intervals, using Mizar system [2], [3], [12]. Differentiation of real one-variable functions is introduced in [13], along standard lines (for interesting survey formalizations analysis various proof-assistants like ACL2 [11], Isabelle/HOL [10], Coq [4], see [5]), but differentiable interval restricted to open intervals. However, when considering relationship with integration [9], since an operation a closed interval, it would be convenient for differentiation able handle derivates as well. Regarding differentiability right and left have already been formalized [6], they are derivatives at endpoints not demonstrated over Therefore, this paper, based these results, although limited functions, we formalize arbitrary intervals summarize them basic propositions. In particular, chain rule [1] important formula relation integration, extending recent results [7], [8] latter field research.