Subset not necessarily Submagma

Theorem
Let $\left({S, \circ}\right)$ be a magma.

Let $T \subseteq S$.

Then it is not necessarily the case that:
 * $\left({T, \circ}\right) \subseteq \left({S, \circ}\right)$

That is, it does not always follow that $\left({T, \circ}\right)$ is a submagma of $\left({S, \circ}\right)$.

Proof
Let $\left({\Z, -}\right)$ be the magma which is the set of integers under the operation of subtraction.

We have that the natural numbers $\N$ are a subset of the integers.

Consider $\left({\N, -}\right)$, the natural numbers under subtraction.

We have that, for example, $1 - 2 = -1 \notin \N$.

Thus $\left({\N, -}\right)$ is not closed.

So $\left({\N, -}\right)$ is not a submagma of $\left({\Z, -}\right)$

Hence it is not true to write $\left({\N, -}\right) \subseteq \left({\Z, -}\right)$, despite the fact that $\N \subseteq \Z$.

Thus $\left({\N, -}\right)$ is not a submagma of $\left({\Z, -}\right)$.