
theorem
for A being non empty closed_interval Subset of REAL,
    D being Division of A,
    rho be Function of A,REAL,
    B be non empty closed_interval Subset of REAL,
    v be FinSequence of REAL
      st B c= A & len D = len v &
         for i be Nat st i in dom v
           holds v.i = vol (B /\ divset(D,i),rho)
 holds Sum v = vol (B,rho)
proof
  let A be non empty closed_interval Subset of REAL,
      D be Division of A,
      rho be Function of A,REAL,
      B be non empty closed_interval Subset of REAL,
      v be FinSequence of REAL;
  assume
AS: B c= A & len D = len v &
    for i be Nat st i in dom v
      holds v.i = vol (B /\ divset(D,i),rho);
  dom rho = A by FUNCT_2:def 1;
  hence Sum v = vol (B,rho) by AS,LmINTEGR208;
end;
