
$$H\Rightarrow\bigwedge_{I^{\prime}\subseteq I}{\bigl(}(\bigwedge_{i\in I^{\prime}}F_{i})\to{\bigl(}\bigvee_{i\in I^{\prime}}F_{i}{\bigr)}{\bigr)}$$

$$\Leftrightarrow\bigwedge_{I^{\prime}\subseteq I}\left(((\bigwedge_{i\in I^{\prime}}F_{i})\land\bigwedge_{i\in I^{\prime}\setminus I}\neg F_{i}\right)\to(\bigvee F_{i})\right)$$

$$\Leftrightarrow(\bigvee_{I^{\prime}\subseteq I}((\bigwedge_{i\in I^{\prime}}F_{i})\wedge\bigwedge_{i\in I^{\prime}\setminus I}\neg F_{i}))\to D.$$

The antecedent of the implication is a tautology: for each interpretation X, the disjunctive term relative to I
′ = {i ∈ I : X |= Fi} is satisfied by X. We can conclude that H entails D.

The proof for antimonotone aggregates is similar.

Proposition 13. *For any aggregate* oph{F1 = w1, . . . , Fn = wn**}i ≺** N*, formula (9) is strongly equivalent to*

$$\bigwedge_{I\subseteq\{1,...,n\}\,:\,o p(\{w_{i}:i\in I\})\not\prec N}\;\;\;i\in\overline{{{I}}}$$

if the aggregate is monotone, and to

$$\bigwedge_{I\subseteq\{1,...,n\}\,:\,o p(\{w_{i}\,:\,i\in I\})\neq N}(\neg\bigwedge_{i\in I}F_{i})$$

if the aggregate is antimonotone.

Proof. Consider the case of a monotone aggregate first. Let G be (9), and H
be (40). In view of Proposition 5, it is sufficient to show that GX is equivalent to HX in classical logic for all sets X. If X 6|= H then also X 6|= G by Lemma 12, so that both reducts are ⊥. Otherwise (X |= H), by the same lemma, X |= G. Then, by Lemma 11, GX is classically equivalent to (39). On the other hand, it is easy to verify, by applying Lemma 2 to HX twice, that HX is classically equivalent to

$$\bigwedge_{I\subseteq\{1,\ldots,n\}\,:\,o p(\{w_{i}:i\in I\})\not\in N}\;\;\;i\in\overline{{{I}}}$$

The claim now follows from Lemma 12.

The reasoning for nonmonotone aggregates is similar.