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 Th5:
  for A be non empty closed_interval Subset of REAL,
      rho be Function of A,REAL,
      u,w be PartFunc of REAL,REAL st
      rho is bounded_variation & dom u = A & dom w = A &
      w = -u & u is_RiemannStieltjes_integrable_with rho holds
        w is_RiemannStieltjes_integrable_with rho &
        integral(w,rho) = -integral(u,rho)
proof
  let A be non empty closed_interval Subset of REAL,
      rho be Function of A,REAL,
      u,w be PartFunc of REAL,REAL;
  assume A1: rho is bounded_variation & dom u = A & dom w = A &
             w = -u & u is_RiemannStieltjes_integrable_with rho; then
A2: w = (-jj)(#)u by VALUED_1:def 6;
  hence w is_RiemannStieltjes_integrable_with rho by A1,Th4;
  integral(w,rho) = (-jj)*integral(u,rho) by A1,A2,Th4;
  hence integral(w,rho) = -integral(u,rho);
end;
