theorem Th41:
  (PartSums(lower_volume(f,D))).(len D) = lower_sum(f,D)
proof
  len lower_volume(f,D) = len D by Def6;
  then len D in Seg(len lower_volume(f,D)) by FINSEQ_1:3;
  then len D in dom lower_volume(f,D) by FINSEQ_1:def 3;
  then
A1: (PartSums(lower_volume(f,D))).(len D) = Sum(lower_volume(f,D)|(len D ))
  by Def19;
  dom lower_volume(f,D)=Seg len lower_volume(f,D) by FINSEQ_1:def 3;
  then dom lower_volume(f,D)=Seg len D by Def6;
  then lower_volume(f,D)|(Seg len D) = lower_volume(f,D);
  hence thesis by A1,FINSEQ_1:def 16;
end;
