Elements of Finite Support form Submagma of Direct Product

Theorem
Let $(S_i,\circ_i)_{i\in I}$ be a family of algebraic structures with unity.

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

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

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

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

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