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 Th34:
  tagged_volume(f + g,TD) = tagged_volume(f,TD) + tagged_volume(g,TD)
  proof
    len tagged_volume(f,TD) = len TD &
    len tagged_volume(g,TD) = len TD by Def4;
    then reconsider R1 = tagged_volume(f,TD),
                    R2 = tagged_volume(g,TD)
                     as Element of len TD-tuples_on REAL by FINSEQ_2:92;
    len tagged_volume(f,TD) = len TD by Def4
                             .= len tagged_volume(g,TD) by Def4; then
Z1: len (tagged_volume(f,TD) + tagged_volume(g,TD))
        = len tagged_volume(f,TD) by RVSUM_1:115
       .= len TD by Def4
       .= len tagged_volume(f + g,TD) by Def4;
    for i be Nat st i in dom tagged_volume(f + g,TD) holds
        (tagged_volume(f + g,TD)).i
        = ((tagged_volume(f,TD)) + tagged_volume(g,TD)).i
    proof
      let i be Nat;
      assume i in dom tagged_volume(f + g,TD);
      then i in Seg len tagged_volume(f + g,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(f + g,TD)).i
        = f.((tagged_of TD).i) * vol(divset(division_of TD,i))
            + g.((tagged_of TD).i) * vol(divset(division_of TD,i)) by Th33
         .= tagged_volume(f,TD).i + g.((tagged_of TD).i)
             * vol(divset(division_of TD,i)) by A1,Def4
         .= R1.i + R2.i by A1,Def4
         .= (tagged_volume(f,TD) + tagged_volume(g,TD)).i by RVSUM_1:11;
      hence thesis;
    end;
    hence thesis by FINSEQ_2:9,Z1;
  end;
