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
  f + g is HK-integrable Function of I,REAL &
  HK-integral (f + g) = HK-integral f + HK-integral g
  proof
    f is HK-integrable;
    then consider J1 being Real such that
A1: for epsilon being Real st epsilon > 0 holds
      ex jauge being positive-yielding Function of I,REAL st
      for TD being tagged_division of I st TD is jauge-fine
      holds |.tagged_sum(f,TD) - J1.| <= epsilon;
    g is HK-integrable;
    then consider J2 being Real such that
A2: for epsilon being Real st epsilon > 0 holds
      ex jauge being positive-yielding Function of I,REAL st
      for TD being tagged_division of I st TD is jauge-fine
      holds |.tagged_sum(g,TD) - J2.| <= epsilon;
A3: HK-integral g = J2 by A2,DEFCC;
A4: for epsilon being Real st epsilon > 0 holds
      ex jauge being positive-yielding Function of I,REAL st
      for TD being tagged_division of I st TD is jauge-fine
      holds |.tagged_sum(f + g,TD) - (J1 + J2).| <= epsilon
    proof
      let epsilon be Real;
      assume
A5:   epsilon > 0;
      set e = epsilon / 2;
      consider jauge1 be positive-yielding Function of I,REAL such that
A6:   for TD being tagged_division of I st TD is jauge1-fine
      holds |.tagged_sum(f,TD) - J1.| <= epsilon / 2 by A5,A1;
      consider jauge2 be positive-yielding Function of I,REAL such that
A7:   for TD being tagged_division of I st TD is jauge2-fine
      holds |.tagged_sum(g,TD) - J2.| <= epsilon / 2 by A5,A2;
      reconsider jauge = min(jauge1,jauge2) as
        positive-yielding Function of I,REAL;
      ex jauge being positive-yielding Function of I,REAL st
      for TD being tagged_division of I st TD is jauge-fine
      holds |.tagged_sum(f + g,TD) - (J1 + J2).| <= epsilon
      proof
        take jauge;
        for TD being tagged_division of I st TD is jauge-fine
          holds |.tagged_sum(f + g,TD) - (J1 + J2).| <= epsilon
        proof
          let TD be tagged_division of I;
          assume TD is jauge-fine;
          then
A8:       TD is jauge1-fine & TD is jauge2-fine by Th23,Th08;
          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;
          tagged_sum(f + g,TD)
            = Sum (tagged_volume(f,TD) + tagged_volume(g,TD)) by Th34
           .= Sum R1 + Sum R2 by RVSUM_1:89
           .= tagged_sum(f,TD) + tagged_sum(g,TD);
          then tagged_sum(f + g,TD) - (J1 + J2)
            = (tagged_sum(f,TD) - J1) + (tagged_sum(g,TD) - J2);
          then |.tagged_sum(f + g,TD) - (J1 + J2).|
            <= |.tagged_sum(f,TD) - J1.| + |.tagged_sum(g,TD) - J2.|
              by COMPLEX1:56;
          then |.tagged_sum(f + g,TD) - (J1 + J2).|
            <= epsilon / 2 + |.tagged_sum(g,TD) - J2.| by A8,A6,Lm01;
          then |.tagged_sum(f + g,TD) - (J1 + J2).|
            <= epsilon / 2 + epsilon / 2 by A8,A7,Lm01;
          hence thesis;
        end;
        hence thesis;
      end;
      hence thesis;
    end;
    then
A9: f + g is HK-integrable;
    then HK-integral(f + g) = J1 + J2 by A4,DEFCC
                           .= (HK-integral f) + (HK-integral g) by A1,DEFCC,A3;
    hence thesis by A9;
  end;
