i = k 1, j 2 1, we have Imf(R ,) I = If(0,t ) f(0,t) I. For

j = 1 1, i 2 k we have m f(Ri ) = (f(si+1,0) f(si,0)(, all as

before. Putting everything together, we have


 m-1
 SImf(R ,j) = If(0,0)I + EZ f(s +1,0) f(s ,0)
 i,j i=k

 n-1
 + E f(O,t ) f(0,tj) + E Ri (f)|
 j=-1 ikk i,j
 jl1


 S|f(0,0)| + Var [,]f(.,0) + Var[,t]f(O,')



 + Var[(oo),(s', (f)](



 = IfI(s',t') = m f (R).


Taking supremum again, we obtain Imf(R) 5S ma f(R). The proof the

other direction is similar to the ones before: for )>0, we choose a

common o,T so that

 ZEf(si+1,0) f(si,0)I > Var 0,sf(.,0) -
 a

 EIf(O,tj+) f(O,t )I > Var tf(0,.) an
 1'],t' 3 and

 E 6R (f)l > Var (f) (
 oT AR ,j [(0,0),(s',t')] 3
 We extend and arbitrarily to partitions ',T of s,s,j
We extend a and arbitrarily to partitions oa,T' of [s,s'%