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This paper presents modeling of a binary tree that represents a natural number and gives an inductive proof for its properties using theorem provers. We define a function for converting data from a natural number into a binary tree and give an inductive proof for its well-definedness. We formalize this method, develop a computational model based on it, and apply it to an electronic cash protocol. We also define the payment function on the binary tree and go on to prove the divisibility of electronic cash using the theorem provers Isabelle/HOL and Coq, respectively. Furthermore, we discuss the effectiveness of this method.