theorem Th44:
  D1 <= D2 & f|A is bounded_below implies lower_sum(f,D2) >=
  lower_sum(f,D1)
proof
  assume that
A1: D1 <= D2 and
A2: f|A is bounded_below;
  len D1 in Seg(len D1) by FINSEQ_1:3;
  then len D1 in dom D1 by FINSEQ_1:def 3;
  then
  (PartSums(lower_volume(f,D1))).(len D1) <= (PartSums(lower_volume(f,D2))
  ).indx(D2,D1,len D1) by A1,A2,Th39;
  then
  lower_sum(f,D1) <= (PartSums(lower_volume(f,D2))).indx(D2,D1,len D1) by Th41;
  then lower_sum(f,D1) <= (PartSums(lower_volume(f,D2))).(len D2) by A1,Th42;
  hence thesis by Th41;
end;
