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 Th32:
  tagged_volume(r(#)f,TD) = r * tagged_volume(f,TD)
  proof
Z1: len (r * tagged_volume(f,TD)) = len tagged_volume(f,TD)
      by RVSUM_1:117
                                        .= len TD by Def4
                                        .= len tagged_volume(r(#)f,TD) by Def4;
    for i be Nat st i in dom tagged_volume(r(#)f,TD) holds
        (tagged_volume(r(#)f,TD)).i = (r * tagged_volume(f,TD)).i
      proof
        let i be Nat;
        assume i in dom tagged_volume(r(#)f,TD);
        then i in Seg len tagged_volume(r(#)f,TD) by FINSEQ_1:def 3;
        then i in Seg len TD by Def4; then
A1:     i in dom TD by FINSEQ_1:def 3;
        then (tagged_volume(r(#)f,TD)).i
          = r * f.((tagged_of TD).i) * vol(divset(division_of TD,i)) by Th31
         .= r * (f.((tagged_of TD).i) * vol(divset(division_of TD,i)))
         .= r * (tagged_volume(f,TD)).i by A1,Def4;
        hence thesis by RVSUM_1:45;
      end;
    hence thesis by FINSEQ_2:9,Z1;
  end;
