As in expression (F.2), for each j ∈ I$_{0}$ and r ∈ { 1 , . . . , k } one has: Using this equality we obtain: Since ∪ k r $_{=1}$C ( j ) r is a union of disjoint events, we obtain for each j ∈ I$_{0}$ and l ∈ { 1 , . . . , k } : Therefore for each j ∈ I$_{0}$ we obtain: k ∑ l =1 1 l k ∑ r =1 ˜ p$_{jrl}$ ≤ k ∑ l =1 1 l [ Pr ( P$_{1}$$_{j}$ ≤ $^{l}$˜ q$_{1}$ m ) − Pr ( P$_{1}$$_{j}$ ≤ ( l − $^{1)}$˜ q$_{1}$ m )] = k l =1 1 l Pr ( P$_{1}$$_{j}$ ≤ $^{l}$˜ q$_{1}$ m ) − k − 1 ∑ l =0 1 l + 1 Pr ( P$_{1}$$_{j}$ ≤ $^{l}$˜ q$_{1}$ m ) = k − 1 ∑ l =1 ( 1 l − 1 l + 1 ) Pr ( P$_{1}$$_{j}$ ≤ $^{l}$˜ q$_{1}$ m ) + 1 k Pr ( P$_{1}$$_{j}$ ≤ $^{k}$˜ q$_{1}$ m ) ≤ k − 1 l =1 1 l + 1 ( ˜ q$_{1}$ m + ˜ q$_{1}$ m = ( ˜ q$_{1}$ m ) k ∑ l =1 1 l . (F.13) k ∑ r =1 ˜ p$_{jrl}$ = Pr ( P$_{1}$$_{j}$ ∈ ( ( l − $^{1)}$˜ q$_{1}$ m , $^{l}$˜ q$_{1}$ m ] , ∪ k r $_{=1}$C ( j ) r ) ≤ Pr P$_{1}$$_{j}$ ∈ ( l − q$_{1}$ m , q$_{1}$ m ]) = Pr ( P$_{1}$$_{j}$ ≤ $^{l}$˜ q$_{1}$ m ) − Pr ( P$_{1}$$_{j}$ ≤ ( l − $^{1)}$˜ q$_{1}$ m ) . ∑ j ∈ I$_{0}$ k ∑ r =1 1 r Pr ( P$_{1}$$_{j}$ ≤ $^{r}$˜ q$_{1}$ m , C ( j ) r ) = ∑ j ∈ I$_{0}$ k ∑ r =1 r ∑ l =1 1 r ˜ p$_{jrl}$ = ∑ j ∈ I$_{0}$ k ∑ l =1 k ∑ r = l 1 r ˜ p$_{jrl}$ ≤ j ∈ I$_{0}$ k l =1 k r = l 1 l p$_{jrl}$ ≤ j ∈ I$_{0}$ k l =1 1 l k r =1 p$_{jrl}$. (F.12) Pr ( P$_{1}$$_{j}$ ≤ $^{r}$˜ q$_{1}$ m , C ( j ) r ) = r ∑ l =1 ˜ p$_{jrl}$. 43