Magma is Submagma of Itself

Theorem
Let $\struct {S, \circ}$ be a magma.

Then $\struct {S, \circ}$ is a submagma of itself.

Proof
By definition, a magma is an algebraic structure $\struct {S, \circ}$ where $\circ$ is closed.

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

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

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

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