theorem Th38:
  D1 <= D2 & i in dom D1 & f|A is bounded_above implies (PartSums(
  upper_volume(f,D1))).i >= (PartSums(upper_volume(f,D2))).indx(D2,D1,i)
proof
  assume that
A1: D1 <= D2 and
A2: i in dom D1 and
A3: f|A is bounded_above;
A4: len upper_volume(f,D2)=len D2 by Def5;
  i in Seg(len D1) by A2,FINSEQ_1:def 3;
  then i in Seg(len upper_volume(f,D1)) by Def5;
  then i in dom upper_volume(f,D1) by FINSEQ_1:def 3;
  then
A5: (PartSums(upper_volume(f,D1))).i=Sum(upper_volume(f,D1)|i) by Def19;
  indx(D2,D1,i) in dom D2 by A1,A2,Def18;
  then indx(D2,D1,i) in Seg(len upper_volume(f,D2)) by A4,FINSEQ_1:def 3;
  then
A6: indx(D2,D1,i) in dom upper_volume(f,D2) by FINSEQ_1:def 3;
  i in Seg(len D1) by A2,FINSEQ_1:def 3;
  then i is non zero Element of NAT by FINSEQ_1:1;
  then
  (PartSums(upper_volume(f,D1))).i >= Sum(upper_volume(f,D2)|indx(D2,D1,i
  )) by A1,A2,A3,A5,Th36;
  hence thesis by A6,Def19;
end;
