reserve a,b,c,d,e for Real;
reserve X,Y for set,
          Z for non empty set,
          r for Real,
          s for ExtReal,
          A for Subset of REAL,
          f for real-valued Function;
reserve I for non empty closed_interval Subset of REAL,
       TD for tagged_division of I,
        D for Division of I,
        T for Element of set_of_tagged_Division(D),
        f for PartFunc of I,REAL;
reserve f for Function of I,REAL;
reserve f,g for HK-integrable Function of I,REAL,
          r for Real;

theorem Th31:
  for i being Nat st i in dom TD holds
  (tagged_volume(r(#)f,TD)).i
    = r * f.((tagged_of TD).i) * vol(divset(division_of TD,i))
  proof
    let i be Nat;
    assume i in dom TD;
    then (tagged_volume(r(#)f,TD)).i
      = (r (#) f).((tagged_of TD).i) * vol(divset(division_of TD,i)) by Def4;
    hence thesis by VALUED_1:6;
  end;
