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 Th35:
  f is HK-integrable implies
  r (#) f is HK-integrable Function of I,REAL &
  HK-integral (r (#) f) = r * HK-integral f
  proof
    assume f is HK-integrable;
    then consider J 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) - J.| <= epsilon;
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(r (#) f,TD) - (r * J).| <= epsilon
    proof
      per cases;
      suppose
A3:     r = 0;
        let epsilon be Real;
        assume
A4:     epsilon > 0;
        set jauge = the positive-yielding Function of I,REAL;
        take jauge;
        for TD be tagged_division of I st TD is jauge-fine
        holds |.tagged_sum(r (#) f,TD) - (r * J).| <= epsilon
        proof
          let TD be tagged_division of I;
          assume TD is jauge-fine;
          tagged_sum(r (#) f,TD) = Sum (r * tagged_volume(f,TD)) by Th32
                                .= r * Sum tagged_volume(f,TD) by RVSUM_1:87
                                .= 0 by A3;
          hence |.tagged_sum(r (#) f,TD) - (r * J).| <= epsilon by A3,A4;
        end;
        hence thesis;
      end;
      suppose
A5:     r <> 0;
        let epsilon be Real;
        assume
A6:     epsilon > 0;
        set e = epsilon / |. r .|;
        consider jauge be positive-yielding Function of I,REAL such that
A7:     for TD be tagged_division of I st TD is jauge-fine
          holds |.tagged_sum(f,TD) - J.| <= e by A1,A5,A6;
        take jauge;
        for TD being tagged_division of I st TD is jauge-fine
          holds |.tagged_sum(r (#) f,TD) - r * J.| <= epsilon
        proof
          let TD be tagged_division of I;
          assume
A8:       TD is jauge-fine;
          |. r * tagged_sum(f,TD) - r * J.|
               = |. r * (tagged_sum(f,TD) - J).|
              .= |.r.| * |. tagged_sum(f,TD) - J.| by COMPLEX1:65;
          then z1: |. r * tagged_sum(f,TD) - r * J.| <= |.r.| * e
             by A7,A8,XREAL_1:64;
          tagged_sum(r (#) f,TD)
                = Sum (r * tagged_volume(f,TD)) by Th32
               .= r * tagged_sum(f,TD) by RVSUM_1:87;
          hence thesis by z1,A5,XCMPLX_1:87;
          end;
          hence thesis;
        end;
      end;
      then
A9:   r (#) f is HK-integrable;
      then HK-integral (r (#) f) = r * J by A2,DEFCC
                                .= r * HK-integral f by A1,DEFCC;
      hence thesis by A9;
    end;
