Restriction of Canonical Surjection to Restricted Dipper Semigroup is Isomorphism

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

Let $+_{m, n}$ be 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:
 * $m + k n \le a + b$

The $+^*_{m, n}$ be the restricted dipper operation on $\N^*_{< \paren {m \mathop + n} }$ defined as:
 * $\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:
 * $m + k n \le a + b$

Consider the algebraic structure $\struct {\N^*_{< \paren {m \mathop + n} }, +^*_{m, n} }$.

Let $\phi^*_{m, n}$ be the restriction of the canonical surjection from $\N_{>0}$ to the restricted dipper semigroup $\struct {\map {D^*} {m, n}, \oplus^*_{m, n} }$.

Then $\phi^*_{m, n}$ is an isomorphism from $\struct {\N^*_{< \paren {m \mathop + n} }, +^*_{m, n} }$ to $\struct {\map {D^*} {m, n}, \oplus^*_{m, n} }$.