reserve s1,s2,q1 for Real_Sequence;
reserve n for Element of NAT;
reserve a,b for Real;

theorem Th6:
  for A be non empty closed_interval Subset of REAL,
      D be Division of A, q be FinSequence of REAL
   st dom q = Seg len D
    & for i be Nat st i in Seg len D holds q.i = vol divset(D,i)
   holds Sum q = vol A
proof
  let A be non empty closed_interval Subset of REAL,
      D be Division of A, q be FinSequence of REAL;
  assume
A1: dom q = Seg len D
   & for i be Nat st i in Seg len D holds q.i = vol divset(D,i);
   set p = lower_volume(chi(A,A),D);

   dom q = Seg len D by A1
        .= Seg len p by INTEGRA1:def 7; then
A2:dom q = dom p by FINSEQ_1:def 3;
   for k be Nat st k in dom q holds q.k = p.k
   proof
    let k be Nat;
    assume A3: k in dom q; then
A4: q.k = vol divset(D,k) by A1;
    k in dom D by A3,A1,FINSEQ_1:def 3;
    hence thesis by A4,INTEGRA1:19;
   end;
   then q = lower_volume ((chi (A,A)),D) by A2,FINSEQ_1:13;
   hence thesis by INTEGRA1:23;
end;
