reserve a,b,e,r,x,y for Real,
  i,j,k,n,m for Element of NAT,
  x1 for set,
  p,q for FinSequence of REAL,
  A for non empty closed_interval Subset of REAL,
  D,D1,D2 for Division of A,
  f,g for Function of A,REAL,
  T for DivSequence of A;

theorem Th9:
  delta(D) >= 0
proof
  consider y being Element of REAL such that
A1: y in rng D by SUBSET_1:4;
  consider n such that
A2: n in dom D and
  y=D.n by A1,PARTFUN1:3;
  n in Seg len D by A2,FINSEQ_1:def 3;
  then n in Seg len upper_volume(chi(A,A),D) by INTEGRA1:def 6;
  then n in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3;
  then upper_volume(chi(A,A),D).n in rng upper_volume(chi(A,A),D) by
FUNCT_1:def 3;
  then
A3: upper_volume(chi(A,A),D).n <= max rng upper_volume(chi(A,A),D) by
XXREAL_2:def 8;
  vol(divset(D,n))=upper_volume(chi(A,A),D).n by A2,INTEGRA1:20;
  then upper_volume(chi(A,A),D).n >= 0 by INTEGRA1:9;
  hence thesis by A3;
end;
