$(E \cup F) = E \cup (F \cap \overline{E})$

These are disjoint so

$u(E \cup F) =u(E) + u(F \cap \overline{E})$

$u(E \cup F) + u(E \cap F) = u(E) + u(F \cap \overline{E}) + u(E \cap F)$

$F=(F \cap E) \cup (F \cap \overline{E})$

and both of these are disjoint so

$u(F) = u(F \cap \overline{E}) + u(E \cap F)$

so

$u(E \cup F) + u(E \cap F)=u(E) + u(F)$