
theorem
  for A be non empty closed_interval Subset of REAL,
    rho be Function of A,REAL,
    u be continuous PartFunc of REAL,REAL st
    rho is bounded_variation & dom u = A
  holds u is_RiemannStieltjes_integrable_with rho
  proof
    let A be non empty closed_interval Subset of REAL,
        rho be Function of A,REAL,
        u be continuous PartFunc of REAL,REAL;
    assume
    A1: rho is bounded_variation & dom u = A;
    A2: u|A is uniformly_continuous by A1,FCONT_2:11;
    consider T0 being DivSequence of A such that
    A3: delta T0 is convergent & lim delta T0 = 0 by INTEGRA4:11;
    set S0 = the middle_volume_Sequence of rho,u,T0;
    set I = lim middle_sum(S0);
    for T being DivSequence of A,
        S be middle_volume_Sequence of rho,u,T
    st delta T is convergent & lim delta T = 0
    holds middle_sum(S) is convergent & lim middle_sum(S) = I
    proof
      let T be DivSequence of A, S be middle_volume_Sequence of rho,u,T;
      assume
      A4: delta T is convergent & lim delta T = 0;
      hence middle_sum(S) is convergent by A1,A2,Th20;
      consider T1 be DivSequence of A such that
      A5: for i be Nat holds
          T1.(2*i) = T0.i & T1.(2*i+1) = T.i by INTEGR20:15;
      consider S1 be middle_volume_Sequence of rho,u,T1 such that
      A6: for i be Nat holds S1.(2*i) = S0.i & S1.(2*i+1) = S.i
            by A5,Th21;
      delta T1 is convergent & lim delta T1 = 0 by A3,A4,A5,INTEGR20:16; then
      A7: middle_sum(S1) is convergent by A1,A2,Th20;
      A8: for i be Nat holds
            (middle_sum(S1)).(2*i) = (middle_sum(S0)).i
          & (middle_sum(S1)).(2*i+1) = (middle_sum(S)).i
      proof
        let i be Nat;
        reconsider S1 as middle_volume_Sequence of rho,u,T1;
        S1.(2*i) = S0.i & T1.(2*i) = T0.i
          & S1.(2*i+1) = S.i & T1.(2*i+1) = T.i by A5,A6; then
        (middle_sum(S1)).(2*i) = Sum(S0.i)
        & (middle_sum(S1)).(2*i+1) = Sum(S.i)
           by INTEGR22:def 7;
        hence thesis by INTEGR22:def 7;
      end;
      lim middle_sum(S) = lim middle_sum(S1) by A7,A8,Th22;
      hence lim middle_sum(S) = lim middle_sum(S0) by A7,A8,Th22;
    end;
    hence u is_RiemannStieltjes_integrable_with rho by INTEGR22:def 8;
  end;
