
theorem
  for A be non empty closed_interval Subset of REAL,
      rho be Function of A,REAL
   st rho is bounded_variation holds
   0 <= total_vd(rho)
proof
  let A be non empty closed_interval Subset of REAL,
      rho be Function of A,REAL;
  assume rho is bounded_variation; then
  consider VD be non empty Subset of REAL such that
A1: VD is bounded_above and
A2: VD = { r where r is Real:
             ex t be Division of A, F be var_volume of rho,t st
               r = Sum(F) } and
A3: total_vd(rho) = upper_bound VD by DeftotalVD;
  reconsider p0=0 as Real;
  for p be ExtReal st p in VD holds p0 <= p
  proof
    let p be ExtReal;
    assume p in VD; then
    consider r be Real such that
B1:   p=r and
B2:   ex t be Division of A, F be var_volume of rho,t st
        r = Sum(F) by A2;
    ex t be Division of A, F be var_volume of rho,t st
    r = Sum(F) by B2;
    hence p0 <= p by B1,LM2;
  end; then
  p0 is LowerBound of VD by XXREAL_2:def 2; then
B4: VD is bounded_below by XXREAL_2:def 9;
B5: for s be Real st s in VD holds 0 <= s
   proof
     let s be Real;
     assume s in VD; then
     consider r be Real such that
B1:    s=r and
B2:    ex t be Division of A, F be var_volume of rho,t st
         r = Sum(F) by A2;
     ex t be Division of A, F be var_volume of rho,t st
     r = Sum(F) by B2;
     hence 0 <= s by B1,LM2;
  end;
  lower_bound VD <= upper_bound VD by A1,B4,SEQ_4:11;
  hence 0 <= total_vd(rho) by A3,SEQ_4:43,B5;
end;
