reserve a,a1,b,b1,x,y for Real,
  F,G,H for FinSequence of REAL,
  i,j,k,n,m for Element of NAT,
  I for Subset of REAL,
  X for non empty set,
  x1,R,s for set;
reserve A for non empty closed_interval Subset of REAL;
reserve A, B for non empty closed_interval Subset of REAL;
reserve r for Real;
reserve D, D1, D2 for Division of A;
reserve f, g for Function of A,REAL;

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;
