it holds Pr Mo B ; C and Pr B 0. By Lemma C.6 b), we get Pr L C)) and
 (1 + min(u =v u ; x =y x
u x  (1 + Pr BL G L H))=Pr B
Pr BL G L H))=Pr B
Pr BL C))

Pr L C))

 u + Pr BL G L H))
Pr B

Pr BL C))

Pr B :
c) Let C GH By Theorem 4.3 b), there exist Pr Mo B ; G and Pr Mo B ; H with
Pr B 0, Pr B 0, u Pr B Pr BL G)), and Pr B Pr BL H)). By

Lemma 3.1, we can choose Pr and Pr such that Pr B Pr B and Pr B L G))

Pr BL H). By Lemmata A.1 and C.6 b), we can choose the probabilistic interpretation

Pr over fB ; L G ; L H g with Pr A Pr A and Pr A Pr A for all atomic
events A and A over fB ; L G g and fB ; L H g respectively, such that:
Pr BL G L H)) max(0; Pr BL H Pr B L G)))
Pr BL G L H)) max(0; Pr BL H)) Pr BL G))) :
By Lemma A.2 with B B B ; G n fBg B B B ; H n fBg B B B L G), and
B L H), there exists probabilistic interpretation Pr over B B ; C with (12). Hence,
it holds Pr Mo B ; C and Pr B 0. By Lemma C.6 b), we get Pr L C)) and

Pr BL G L H)) Pr L G L H)) Pr BL C)) Pr L C))

 Pr BL G L H)) Pr B)) Pr BL C)) Pr B :

d) Let C GH By Theorem 3.2, there is model Pr 000 Mo B ; C with Pr 000 BL C)) 0.
By Theorem 4.3 b), there is model Pr 00 Mo B ; C with Pr 00 B and Pr 00 B
 Pr 00 B Pr 00 BL C)). Hence, there is model Pr 0 Mo B ; C with Pr 0B and
min("; Pr 000 BL C)) Pr 000 B)) Pr 0B Pr 0BL C)) :
Let the models Pr Mo B ; G and Pr Mo B ; H be dened by Pr A Pr 0A and
Pr A Pr 0A for all atomic events A and A over B B ; G and B B ; H), respectively.
By Lemma 3.1, we can choose Pr and Pr such that Pr B Pr B and Pr B L G))

Pr BL H). By Lemmata A.1 and C.6 b), we can choose the probabilistic interpretation

Pr over fB ; L G ; L H g with Pr A Pr A and Pr A Pr A for all atomic
events A and A over fB ; L G g and fB ; L H g respectively, such that:
Pr BL G L H)) max(0; Pr BL H Pr B L G)))
Pr BL G L H)) min("; Pr 000 BL C)) Pr 000 B)) Pr B :
By Lemma A.2 with B B B ; G fBg B B B ; H fBg B B B L G), and
B L H), there is probabilistic interpretation Pr over B B ; C with (12). Hence, it holds
Pr Mo B ; C), Pr B 0, Pr L C)) 0, Pr BL C)) = 0, and " Pr B Pr BL C )). 2
  **Proof of Theorem 4.8.**$u$  $u$  $E$  $Pr$$E$  $Pr$$KB$  $Pr$$KB$  $Pr$$KB$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$E$  $Pr$$  $Pr$$E$  $Pr$$  $Pr$$E$  

