Composition of Cartesian Products of Mappings

Theorem
Let $I$ be an indexing set.

Let $\family {S_\alpha}_{\alpha \mathop \in I}$, $\family {T_\alpha}_{\alpha \mathop \in I}$ and $\family {U_\alpha}_{\alpha \mathop \in I}$ be families of sets all indexed by $I$.

For each $\alpha \in I$, let:
 * $f_\alpha: S_\alpha \to T_\alpha$ be a mapping
 * $g_\alpha: T_\alpha \to U_\alpha$ be a mapping.

Let $\ds f: \prod_{\alpha \mathop \in I} S_\alpha \to \prod_{\alpha \mathop \in I} T_\alpha$ and $\ds g: \prod_{\alpha \mathop \in I} T_\alpha \to \prod_{\alpha \mathop \in I} U_\alpha$ be defined as:


 * $\ds f = \prod_{\alpha \mathop \in I} f_\alpha$
 * $\ds g = \prod_{\alpha \mathop \in I} g_\alpha$

Then their composition $\ds g \circ f: \prod_{\alpha \mathop \in I} S_\alpha \to \prod_{\alpha \mathop \in I} U_\alpha$ is:
 * $\ds g \circ f: \prod_{\alpha \mathop \in I} g_\alpha \circ f_\alpha$