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 Th22:
  Sum(upper_volume(chi(A,A),D))=vol(A)
proof
A1: for i be Nat st 1 <= i & i <= len(lower_volume(chi(A,A),D)) holds
  lower_volume(chi(A,A),D).i=upper_volume(chi(A,A),D).i
  proof
    let i be Nat;
    assume that
A2: 1 <= i and
A3: i <= len(lower_volume(chi(A,A),D));
    i <= len D by A3,Def6;
    then
A4: i in dom D by A2,FINSEQ_3:25;
    then lower_volume(chi(A,A),D).i=vol(divset(D,i)) by Th17
      .=upper_volume(chi(A,A),D).i by A4,Th18;
    hence thesis;
  end;
  len (lower_volume(chi(A,A),D)) = len D by Def6
    .= len (upper_volume(chi(A,A),D)) by Def5;
  then lower_volume(chi(A,A),D)=upper_volume(chi(A,A),D) by A1,FINSEQ_1:14;
  hence thesis by Th21;
end;
