Identity of Subgroup of Big Dipper Semigroup is not Identity of Big Dipper

Theorem
Let $m, n \in \Z$ be integers such that $m, n > 0$.

Let $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$ denote the big dipper semigroup.

Let $\struct {H, +_{m, n} }$ be the subgroup of $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$ where $H = \set {k \in \N: m \le k < m + n}$

Then the identity of $\struct {H, +_{m, n} }$ is not the identity of $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$.

Proof
This is demonstrated by Proof by Counterexample.

First we note that by Existence of Subgroup of Big Dipper Semigroup:
 * $\struct {H, +_{m, n} }$ is indeed a subgroup of $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$
 * the identity of $\struct {H, +_{m, n} }$ is $n$.

But we note that by definition of $+_{m, n}$:
 * $0 +_{m, n} n = n - k n$

where $m \le n - k n$

As we have specified that $m > 0$, it follows that:
 * $0 +_{m, n} n > 0$

Thus $n$ is not the identity of $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$.