bounded by q$_{2}$. Combining this result with (F.18), we obtain FDR ≤ q$_{1}$ + q$_{2}$ = q$_{1}$ + q − q$_{1}$ = q. F.1 Proof of Lemma F.1 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: 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}$$_{i}$ for i ∈ R ( j ) 1 ( p ( j ) 1 ) . For a given p$_{1}$ and j ∈ I$_{10}$ ∩ R $_{1}$( p$_{1}$ ), using the fact that for each s ∈ { 1 , . . . , m − 1 } , D ( j ) s ∩ { P$_{1}$ = p$_{1}$ } is an increasing set for P ( j ) 2 , as well as the PRDS property of the p -values from the follow-up study and the independence of the p -values across the studies, we obtain for each s ∈ { 1 , . . . , R$_{1}$ ( p$_{1}$ ) − 1 } : Using the fact that for each s ∈ { 1 , . . . , R$_{1}$ ( p$_{1}$ ) − 1 } , D ( j ) s ∪ C ( j ) s +1 = D ( j ) s $_{+1}$, where D ( j ) s Pr ( D ( j ) s | P$_{2}$$_{j}$ ≤ sq$_{2}$ R$_{1}$ ( p$_{1}$ ) , P$_{1}$ = p$_{1}$ ) ≤ Pr ( D ( j ) s | P$_{2}$$_{j}$ ≤ ( s + 1) q$_{2}$ R$_{1}$ ( p$_{1}$ ) , P$_{1}$ = p$_{1}$ ) . (F.19) D ( j ) s = { ( P ( j ) 1 , P ( j ) 2 ) : T$_{(}$$_{s}$$_{)}$ > s + 1 , T$_{(}$$_{s}$$_{+1)}$ > s + 2 , . . . , T$_{(}$$_{m}$$_{−}$$_{1)}$ > m } , 45