Dilation of Subset of Vector Space Distributes over Sum/General Case

Theorem
Let $K$ be a field.

Let $X$ be a vector space over $K$. Let $\family {E_\alpha}_{\alpha \mathop \in A}$ be an $A$-indexed family of sets.

Let $\lambda \in K$.

Then:
 * $\ds \lambda \sum_{\alpha \mathop \in A} E_\alpha = \sum_{\alpha \mathop \in A} \paren {\lambda E_\alpha}$

Proof
Let $x \in X$.

We have:
 * $\ds x \in \lambda \sum_{\alpha \mathop \in A} E_\alpha$

there exists:
 * a finite subset $F \subseteq E_\alpha$
 * $x_\alpha \in F$ for each $\alpha \in F$

such that:
 * $\ds x = \lambda \sum_{\alpha \in F} x_\alpha$

This is equivalent to:
 * $\ds x = \sum_{\alpha \in F} \lambda x_\alpha$

for a finite subset $F \subseteq E_\alpha$, with $x_\alpha \in F$ for each $\alpha \in F$.

Hence we obtain:
 * $\ds x \in \lambda \sum_{\alpha \mathop \in A} E_\alpha$ $x \in \sum_{\alpha \mathop \in A} \paren {\lambda E_\alpha}$

so that:
 * $\ds \lambda \sum_{\alpha \mathop \in A} E_\alpha = \sum_{\alpha \mathop \in A} \paren {\lambda E_\alpha}$