Restricted Dipper Operation is Associative

Theorem
Let $m, n \in \N_{>0}$ be non-zero natural numbers.

Let $\N_{< \paren {m \mathop + n} }$ denote the initial segment of the natural numbers:
 * $\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m + n - 1}$

Let $\N^*_{< \paren {m \mathop + n} }$ denote the set defined as $\N_{< \paren {m \mathop + n} } \setminus \set 0$:
 * $\N^*_{< \paren {m \mathop + n} } := \set {1, 2, \ldots, m + n - 1}$

The restricted dipper operation $+^*_{m, n}$ on $\N^*_{< \paren {m \mathop + n} }$ is associative.

Proof
By definition, $+^*_{m, n}$ is the restriction of the dipper relation $+_{m, n}$ to $\N_{>0}$.

We have from Dipper Operation is Associative that $+_{m, n}$ is an associative operation.

The result follows from Restriction of Associative Operation is Associative.

Also see

 * Restricted Dipper Operation is Commutative