reserve A for non empty closed_interval Subset of REAL;
reserve rho for Function of A,REAL;
reserve u for PartFunc of REAL,REAL;
reserve T for DivSequence of A;
reserve S for middle_volume_Sequence of rho,u,T;
reserve k for Nat;

theorem
  for A be non empty closed_interval Subset of REAL,
      rho be Function of A,REAL,
      u,v,w be PartFunc of REAL,REAL st
      rho is bounded_variation & dom u = A & dom v = A & dom w = A &
      w = u - v & u is_RiemannStieltjes_integrable_with rho &
      v is_RiemannStieltjes_integrable_with rho holds
        w is_RiemannStieltjes_integrable_with rho &
        integral(w,rho) = integral(u,rho) - integral(v,rho)
proof
  let A be non empty closed_interval Subset of REAL,
      rho be Function of A,REAL,
      u,v,w be PartFunc of REAL,REAL;
  assume A1: rho is bounded_variation &
             dom u = A & dom v = A & dom w = A &
             w = u - v & u is_RiemannStieltjes_integrable_with rho &
             v is_RiemannStieltjes_integrable_with rho; then
A2: w = u + (-v) by VALUED_1:def 9;
A3: dom (-v) = A by A1,VALUED_1:8;
  reconsider gg = -v as PartFunc of REAL,REAL;
A5: gg is_RiemannStieltjes_integrable_with rho by A1,Th5,A3;
  hence w is_RiemannStieltjes_integrable_with rho by A1,A2,Th6,A3;
  integral(w,rho) = integral(u,rho) + integral(gg,rho) by A1,A2,A5,Th6,A3; then
  integral(w,rho) = integral(u,rho) + -integral(v,rho) by A1,Th5,A3;
  hence integral(w,rho) = integral(u,rho) - integral(v,rho);
end;
