Restriction of Dipper Operation to Non-Zero Initial Segment is Closed

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 $+_{m, n}$ denote the dipper operation on $\N_{< \paren {m \mathop + n} }$:
 * $\forall a, b \in \N_{< \paren {m \mathop + n} }: a +_{m, n} b = \begin{cases}

a + b & : a + b < m \\ a + b - k n & : a + b \ge m \end{cases}$ where $k$ is the largest integer satisfying:
 * $m + k n \le a + b$

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

Then $+_{m, n}$ is closed on $\N^*_{< \paren {m \mathop + n} }$.

Proof
From Dipper Operation is Closed on Initial Segment, $+_{m, n}$ is closed on $\N_{< \paren {m \mathop + n} }$.

That is:
 * $\forall a, b \in \N_{< \paren {m \mathop + n} }: a +_{m, n} b < m + n$

It remains to be shown that:
 * $\forall a, b \in \N^*_{< \paren {m \mathop + n} }: a +_{m, n} b > 0$

Let $a + b < m$.

Then as $a > 0$ and $b > 0$ it follows that $a + b < 0$.

That is:
 * $a +_{m, n} b > 0$

Let $a + b > m$.

Then:
 * $a +_{m, n} b = a + b - k n$

where $k$ is the largest integer satisfying $m + k n \le a + b$.

That is:
 * $m < a + b - k n$

But as $m > 0$ we must have that:
 * $0 < a +_{m, n} b$

and the result follows.