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,rho1 be Function of A,REAL,
      u be PartFunc of REAL,REAL st
      rho is bounded_variation & rho1 is bounded_variation & dom u = A &
      rho = -rho1 & u is_RiemannStieltjes_integrable_with rho1 holds
        u is_RiemannStieltjes_integrable_with rho &
        integral(u,rho) = -integral(u,rho1)
proof
  let A be non empty closed_interval Subset of REAL,
      rho,rho1 be Function of A,REAL,
      u be PartFunc of REAL,REAL;
  assume A1: rho is bounded_variation & rho1 is bounded_variation & dom u = A &
             rho = -rho1 & u is_RiemannStieltjes_integrable_with rho1; then
A2: rho = (-jj)(#)rho1 by VALUED_1:def 6;
  hence u is_RiemannStieltjes_integrable_with rho by A1,Th4A;
  integral(u,rho) = (-jj)*integral(u,rho1) by A1,A2,Th4A;
  hence integral(u,rho) = -integral(u,rho1);
end;
