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 Th39:
  D1 <= D2 & i in dom D1 & f|A is bounded_below implies (PartSums(
  lower_volume(f,D1))).i <= (PartSums(lower_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_below;
A4: len lower_volume(f,D2)=len D2 by Def6;
  i in Seg(len D1) by A2,FINSEQ_1:def 3;
  then i in Seg(len lower_volume(f,D1)) by Def6;
  then i in dom lower_volume(f,D1) by FINSEQ_1:def 3;
  then
A5: (PartSums(lower_volume(f,D1))).i=Sum(lower_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 lower_volume(f,D2)) by A4,FINSEQ_1:def 3;
  then
A6: indx(D2,D1,i) in dom lower_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(lower_volume(f,D1))).i <= Sum(lower_volume(f,D2)|indx(D2,D1,i
  )) by A1,A2,A3,A5,Th37;
  hence thesis by A6,Def19;
end;
