
theorem
 for A be non empty closed_interval Subset of REAL,
     rho be Function of A,REAL,
     T be Division of A
   st rho is bounded_variation holds
   for F be var_volume of rho,T
     holds Sum(F) <= total_vd(rho)
proof
  let A be non empty closed_interval Subset of REAL,
      rho be Function of A,REAL,
      T be Division of A;
  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;
  let F be var_volume of rho,T;
  Sum(F) in VD by A2;
  hence thesis by A1,SEQ_4:def 1,A3;
end;
