Splitting and Updating Hybrid Knowledge Bases (Extended Version) 33 Furthermore, since φ is subjective, we can use Proposition 49 to further rewrite the previous formula into
(∃I ∈ I)
DI, M[U], N[U]E|= φ which is by Definition 50 equivalent to

$\downarrow$ . 

DM[U], N[U]E|= φ .

Corollary 53 Let φ be a subjective formula, U ⊆ P be a set of predicate symbols such that U ⊇ pr(φ)
and M, M′, N, N′ ∈ M be such that M coincides with M′ on U and N coincides with N′ on U. Then hM, Ni |= φ ⇐⇒ hM′, N′i |= φ .

Proof By assumptions we know that M[U] = M′[U]and N[U] = N′[U]. Proposition 52 now yields:
hM, Ni |= φ ⇐⇒ DM[U], N[U]E|= φ ⇐⇒ DM′[U], N′[U]E|= φ

$$M,N\rangle\models\phi\Longleftrightarrow\left\langle M^{\langle b\rangle},N^{\langle b\rangle}\right\rangle\models\phi\Longleftrightarrow\left\langle M^{\langle b\rangle},N^{\langle b\rangle}\right\rangle\models\phi$$ $$\Longleftrightarrow\left\langle M^{\prime},N^{\prime}\right\rangle\models\phi\;\;.\quad\hbox{\vrule width0.0pt height6.0pt depth0.0pt\vrule width0.0pt height6.0pt depth0.0pt}$$

## Appendix B Proof Of Splitting Set Theorem For Hybrid Mknf Knowledge Bases

Remark 54 Note that whenever U is a splitting set for a hybrid knowledge base K = hO,Pi, the following can be easily shown to hold:
pr(bU (O)) ⊆ U ,
pr(bU (P)) ⊆ U ,
pr(π(bU (K))) ⊆ U ,
pr(tU (O)) ⊆ P \ U .

Also note that the heads of rules in tU (P) contain only predicate symbols from P \ U
while their bodies may also contain predicate symbols from U. However, for any X ∈ M,
the above defined reducts eU (P, X) and eU (K, X) can be shown to mention only atoms not belonging to U:
pr(π(eU (K, X))) ⊆ P \ U .

Please stay warned that these rather basic observations will be used in the following text
(especially in the proofs) without further notice or reference.