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;
