Proof
(related to Proposition: Recursively Defined Arithmetic Functions, Recursion)
 By hypothesis, $N\in\mathbb N$ is a given natural number.
 It is sufficient to show that the arithmetic function $f:\mathbb N\to\mathbb C$ defined by specifying the value of $f(m)$ for all $m\le N$ and for all $n > N$ with a given recursion formula $f(n)=\mathcal R(f(m)\mid m < n)$ is welldefined (i.e. unique) for all natural numbers.
 Assume, $f$ defined like this is not welldefined for some natural numbers.
 Because of the wellordering principle, there is the smallest natural number $N_0\in\mathbb N$ for which $f$ is not welldefined.
 But then for $m < N_0$ the values $f(m)$ are welldefined and $f(N_0)=\mathcal R(f(m)\mid m < N_0).$
 This is a contradiction.
 Therefore, $f$ is welldefined for all natural numbers.
∎
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References
Bibliography
 Aigner, Martin: "Diskrete Mathematik", vieweg studium, 1993