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### Abstract

This work formalizes Zeckendorf's theorem. The theorem states that every positive integer can be uniquely represented as a sum of one or more non-consecutive Fibonacci numbers. More precisely, if $N$ is a positive integer, there exist unique positive integers $c_i \ge 2$ with $c_{i+1} > c_i + 1$, such that
\[
N = \sum_{i=0}^k F_{c_i}
\]
where $F_n$ is the $n$-th Fibonacci number.