content_image/327405.jpg content_image/327413.jpg content_image/327412.jpg
bounded by q$_{2}$. Combining this result with (F.18), we obtain 

## F.1 Proof of Lemma F.1 

<img src='content_image/327405.jpg'>

Proof of item 1. Our proof is similar to the proof of Theorem 1.2 in Benjamini and Yekutieli (2001). For j ∈ { 1 , . . . , m } and s ∈ { 1 , . . . , m − 1 } we define the event D ( j ) s as follows: 

<img src='content_image/327413.jpg'>

and we define D ( j ) m to be the entire sample space of ( P ( j ) 1 , P ( j ) 2 ) . Note that D ( j ) s = ∪ s r $_{=1}$C ( j ) r . It is easy to see that D ( j ) s is the event in which if H$_{NR,j}$ is rejected by Procedure 3.2, at most s hypotheses are rejected including H$_{NR,j}$ . 

We will first show that for each p$_{1}$ , j ∈ I$_{10}$ ∩R $_{1}$( p$_{1}$ ) and s ∈ { 1 , . . . , m − 1 } , D ( j ) s ∩{ P$_{1}$ = p$_{1}$ } is an increasing set for P ( j ) 2 , i.e. if ( P$_{1}$, P ( j ) 2 ) ∈ D ( j ) s ∩ { P$_{1}$ = p$_{1}$ } and ˜ P ( j ) 2 ≥ P ( j ) 2 , then ( P$_{1}$, ˜ P ( j ) 2 ) ∈ D ( j ) s ∩ { P$_{1}$ = p$_{1}$ } . The result follows from the fact that for fixed P$_{1}$ = p$_{1}$ and j ∈ I$_{10}$ ∩ R $_{1}$( p$_{1}$ ) , T$_{i}$ = ∞ for i / ∈ R ( j ) 1 ( p ( j ) 1 ) , and T$_{i}$ is increasing in P$_{2}$$_{i}$ for i ∈ R ( j ) 1 ( p ( j ) 1 ) . 

For a given p$_{1}$ and j ∈ I$_{10}$ ∩ R $_{1}$( p$_{1}$ ), using the fact that for each s ∈ { 1 , . . . , m − 1 } , D ( j ) s ∩ { P$_{1}$ = p$_{1}$ } is an increasing set for P ( j ) 2 , as well as the PRDS property of the p -values from the follow-up study and the independence of the p -values across the studies, we obtain for each s ∈ { 1 , . . . , R$_{1}$ ( p$_{1}$ ) − 1 } : 

<img src='content_image/327412.jpg'>

Using the fact that for each s ∈ { 1 , . . . , R$_{1}$ ( p$_{1}$ ) − 1 } , D ( j ) s ∪ C ( j ) s +1 = D ( j ) s $_{+1}$, where D ( j ) s 

