Magma is Submagma of Itself

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

Then $\left({S, \circ}\right)$ is a submagma of itself.

Proof
By definition, a magma is an algebraic structure $\left({S, \circ}\right)$ where $\circ$ is closed.

That is:
 * $\forall x, y \in S: x \circ y \in S$

From Subset of Itself, $S \subseteq S$.

By definition, $\left({T, \circ}\right)$ is a submagma of $S$ if:
 * $\forall x, y \in T: x \circ y \in T$

It follows from the above that $\left({S, \circ}\right)$ is, by definition, a submagma of itself.