Hence, Pr satises the same conditions. Since L G L H L GH L C and by Lem-

mata A.1, C.5, and C.6 b), we then get:

 (1 + min(u =v u ; x =y x
u x  (1 + Pr BL G L H))=Pr B
$$+\ {\frac{P r\ {\overline{{B L}}}\ G\ L\ H\ \ \ /P r\ B}{P r\ B L\ G\ L\ H\ \ \ /P r\ B}}$$
$+\;\hdots$  . 
Pr BL C))

Pr L C))
:
b)

Basis: Let C B model Pr Mo B ; C such that Pr B 0, Pr L C)) 0, Pr L C)) Pr BL C)), and Pr B Pr BL C)) is given by B ; B ; 1.

Induction: Let C B Let the model Pr of f C B)[u ; u ; B C)[ ; v g with Pr B
and Pr C be dened like in the proof of Theorem 3.2.

By the induction hypothesis, there is Pr Mo C; C"with Pr C 0, Pr L C")) 0, y Pr L C")) Pr CL C")), and Pr C Pr CL C")). By Lemma 3.1, we can choose Pr and Pr such that Pr C Pr C and Pr B C Pr CL C"
)). By Lemmata A.1 and C.6 a), we can choose the probabilistic interpretation Pr over fB ; C; L C"g with Pr A Pr A and Pr A Pr A for all atomic events A and A over fB ; Cg and fC; L C"g respectively, such that:
Pr BCL C"
)) max(0; Pr CL C")) Pr B C))
Pr BCL C "
)) max(0; Pr CL C")) Pr BC)) :
By Lemma A.2 with B fBg B B C; C"nfCg B C B B and B L C"),
there exists probabilistic interpretation Pr over B B ; C with (12). Hence, it holds Pr Mo B ; C), Pr B 0, and Pr L C)) 0. Moreover, by Lemma C.6 a), we get:
 yy x + y v x Pr CL C")) Pr L C")) Pr CL C)) Pr L C))
 u u v + u x v Pr CL C")) Pr C Pr CL C)) Pr C :
Let C GH By the induction hypothesis, there are models Pr Mo B ; G and Pr Mo B ; H with Pr B 0, Pr B 0, Pr L G)) 0, Pr L H)) 0, and u Pr B Pr BL G)); Pr B Pr BL H))
Pr L G)) Pr BL G)); y Pr L H)) Pr BL H)) :
By Lemma 3.1, we can choose Pr and Pr with 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)) min(Pr BL H)); Pr BL 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,