
theorem Th19:
  for A be non empty closed_interval Subset of REAL,
    rho be Function of A,REAL,
    T,S be Division of A
  st rho is bounded_variation holds
   ex ST be FinSequence of REAL
   st len ST = len S
    & Sum ST <= total_vd(rho)
    & for j be Nat st j in dom S holds
      ex p be FinSequence of REAL
      st ST.j = Sum p
       & len p = len T
       & for i be Nat st i in dom T holds
         p.i = |. vol(divset(T,i) /\ divset(S,j),rho) .|
  proof
    let A be non empty closed_interval Subset of REAL,
        rho be Function of A,REAL,
        T,S be Division of A;
    assume
    A1: rho is bounded_variation;
    defpred P[Nat,object] means
    ex p be FinSequence of REAL
    st $2 = Sum p
     & len p = len T
     & for i be Nat st i in dom T
       holds p.i = |. vol(divset(T,i) /\ divset(S,$1),rho) .|;
    A2: for j be Nat st j in Seg len S holds
        ex x be Element of REAL st P[j,x]
    proof
      let j be Nat;
      assume j in Seg len S;
      defpred Q[Nat,object] means
      $2= |. vol (divset(T,$1) /\ divset(S,j),rho) .|;
      A3: for i be Nat st i in Seg len T holds
          ex y be Element of REAL st Q[i,y]
      proof
        let i be Nat;
        assume i in Seg len T;
        |. vol (divset(T,i) /\ divset(S,j),rho) .| in REAL
          by XREAL_0:def 1;
        hence thesis;
      end;
      consider p be FinSequence of REAL such that
      A4: dom p = Seg len T
        & for i be Nat st i in Seg len T
          holds Q[i,p.i] from FINSEQ_1:sch 5(A3);
      reconsider x = Sum p as Element of REAL by XREAL_0:def 1;
Z2:   dom T = Seg len T by FINSEQ_1:def 3;
      len p = len T by A4,FINSEQ_1:def 3; then
      P[j,x] by A4,Z2;
      hence ex x be Element of REAL st P[j,x];
    end;
    consider ST be FinSequence of REAL such that
    A5: dom ST = Seg len S
      & for j be Nat st j in Seg len S
        holds P[j,ST.j] from FINSEQ_1:sch 5(A2);
    take ST;
    thus
    A6: len ST = len S by A5,FINSEQ_1:def 3;
a6: dom ST = dom S by A5,FINSEQ_1:def 3; then
    consider H be Division of A, F be var_volume of rho,H such that
    A7: Sum ST = Sum(F) by A1,A5,A6,Lm2;
    thus thesis by A1,A5,A7,INTEGR22:5,a6;
  end;
