Let $\Bbb S = \left\{{\odot:}\right.$ for some $n \in \N_{>0}$, $\odot$ is an $n$-ary operation on $\left.{S}\right\}$. Let $s: \Bbb S \to \Bbb S$ be the mapping defined as follows. Let $\odot$ be any $n$-ary operation defined on $\Bbb S$. Then $s \left({\odot}\right)$ is the $\left({n+1}\right)$-ary operation defined by: :$\forall \left({a_1, \ldots, a_n, a_{n+1}}\right) \in S^{n+1}: s \left({\odot}\right) \left({a_1, \ldots, a_n, a_{n+1}}\right) = \odot \left({a_1, \ldots, a_n}\right) \oplus a_{n+1}$ By the Principle of Recursive Definition, there is a unique sequence $\left \langle {\oplus_k} \right \rangle_{k \mathop \ge 1}$ such that: : $\oplus_1$ is the unary operation defined as $\oplus_1 \left({a}\right) = a$ and: : $\oplus_{n+1} = s \left({\oplus_n}\right)$ for each $n \in \N_{>0}$. Let $A$ be the set defined as the set of all $n$ such that: :$\oplus_n$ is an $n$-ary operation on $S$ and: :by the definition of $s$, $(2)$ holds for every ordered $n + 1$-tuple in $S^{n+1}$. It will be demonstrated by the Principle of Mathematical Induction that $A = \N$. === Basis for the Induction === We have that: :$\oplus_1$ is the unary operation defined as $\oplus_1 \left({a}\right) = a$ By the Principle of Recursive Definition: :$\oplus_2 = s \left({\oplus_1}\right)$ is the unique operation on $S$ such that: {{begin-eqn}} {{eqn | lo= \forall \left({a_1, a_2}\right) \in S^2:       | l = \oplus_2 \left({a_1, a_2}\right)       | r = s \left({\oplus_1}\right) \left({a_1, a_2}\right)       | c = Definition of $s$ }} {{eqn | r = \oplus_1 \left({a_1}\right) \oplus a_2       | c =  }} {{eqn | r = a_1 \oplus a_2       | c =  }} {{end-eqn}} By the definition of a magma, $\oplus$ is a operation defined such that: :$\forall \left({a_1, a_2}\right) \in S \times S: \oplus \left({a_1, a_2}\right) \in S$ where $\oplus: S \times S \to S$ is a mapping. $a_1 \oplus a_2$ is defined as: :$\forall \left({a_1, a_2}\right) \in S \times S: a_1 \oplus a_2 := \oplus \left({a_1, a_2}\right)$ So $1 \in A$. This is our basis for the induction. === Induction Hypothesis === It is to be shown that, if $m \in A$ where $m \ge 1$, then it follows that $m + 1 \in A$. This is the induction hypothesis: :$\oplus_m$ is an $m$-ary operation on $S$ and: :by the definition of $s$, $(2)$ holds for every ordered $m + 1$-tuple in $S^{m+1}$. It is to be demonstrated that it follows that: :$\oplus_{m+1}$ is an $m+1$-ary operation on $S$ and: :by the definition of $s$, $(2)$ holds for every ordered $m + 2$-tuple in $S^{m+2}$. === Induction Step === This is our induction step: By the Principle of Recursive Definition: :$\oplus_{m+1} = s \left({\oplus_m}\right)$ is the unique operation on $S$ such that: {{begin-eqn}} {{eqn | lo= \forall \left({a_1, a_2, \ldots, a_{m+1} }\right) \in S^{m+1}:       | l = \oplus_{m+1} \left({a_1, a_2, \ldots, a_{m+1} }\right)       | r = s \left({\oplus_m}\right) \left({a_1, a_2, \ldots, a_{m+1} }\right)       | c = Definition of $s$ }} {{eqn | r = \oplus_m \left({a_1, a_2, \ldots, a_m}\right) \oplus a_{m+1}       | c =  }} {{eqn | r = \left({a_1 \oplus a_2 \oplus \cdots \oplus a_m}\right) \oplus a_{m+1}       | c =  }} {{end-eqn}} By the definition of a magma, $\oplus$ is a operation defined such that: :$\forall \left({a_1, a_2}\right) \in S \times S: \oplus \left({a_1, a_2}\right) \in S$ where $\oplus: S \times S \to S$ is a mapping. By the induction hypothesis: :$\oplus_m \left({a_1, a_2, \ldots, a_m}\right) \oplus a_{m+1} \in S$ So by definition: $\oplus_{m+1} \left({a_1, a_2, \ldots, a_m, a_{m+1} }\right) \in S$ Let $b := \oplus_{m+1} \left({a_1, a_2, \ldots, a_m, a_{m+1} }\right)$. But then by the basis of the induction: :$b \oplus a_{m+2} \in S$ That is: :$\oplus_{m+1} \left({a_1, a_2, \ldots, a_m, a_{m+1} }\right) \oplus a_{m+2} = \oplus_{m+2} \left({a_1, a_2, \ldots, a_m, a_{m+1}, a_{m+2} }\right) \in S$ So $m \in A \implies m + 1 \in A$ and the result follows by the Principle of Mathematical Induction: {{Qed}}