(S0,1,rO) (PO,ro)

 I III
 (s 't0, POl) l0,1


 II
 Si -----------'(po,tI)


 t,1

 (s ,t')
 s (sO, ,'t )


 Figure 2-5 Computing variation of R1



variation of f on the respective rectangles, we have



+ V[- ]ar (f)
Var (s ,t ,1),'(s0,1 ,r0)](f) + Var[(s0,1',t'),(p t o,



 + Var (f) > c ,
 P(s ,t1O' 0,1 0r0)]t)I 2


Denote u = (S0,1'taI) = (l ,rl) > z'. By assumption,

Var[z ,u (f) > a, so there exists a partition oa: s' = s1,0 < s1,1

< ... < s1,m p1 1: t' t,0 < t, < ... < t ,n = r1 such that

 E JA f > a and 1r(s t )(s ,t f < Then, as
RBE:co 1 0,0't1,0 1,1

above, we have that the total variation of f on the three rectangles

comprising [(s' ,t'),pr) t'),(s ,t1,1 )] is greater than

a hence the total variation of f on the rectangles comprising

[(s',t'),(p0,rO)] \ [(s',t'),(s ,t, )] > (a ) + (a ) =

2, ( + ).
 2 4