Elements of Finite Support form Submagma of Direct Product

Theorem
Let $\left({S_i, \circ_i}\right)_{i \mathop \in I}$ be a family of magmas with itentity.

Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be their direct product.

Let $T$ be the subset of elements of $S$ whose support is finite:
 * $T = \left\{ {s \in S: \operatorname{supp} \left({s}\right) \text{ is finite} }\right\}$

Then $T$ is a submagma of $S$.

Proof
From Finite Subsets form Ideal, the set of finite subsets of $I$ form an ideal of $I$.

From Elements with Support in Ideal form Submagma of Direct Product, $T$ is a submagma of $S$.