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 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;
