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**p00ndawg** Suppose $\displaystyle Y \in F(A) \cup F(B) $and $\displaystyle y = F(x). $

Since $\displaystyle y \in F(A) \cup F(B) $, $\displaystyle Y \in F(A) $ or $\displaystyle Y \in F(B) $.

If $\displaystyle Y \in F(A) $, then $\displaystyle x \in A $, and if $\displaystyle y \in F(B) $then $\displaystyle x \in B $.

Because $\displaystyle y \in F(A) \cup F(B) $, it follows that $\displaystyle x \in A $ or $\displaystyle x \in B $.

Thus, $\displaystyle F(A) \cup F(B) \subseteq F(A \cup B) $.