reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  c for Complex,
  E,A,B for Element of S;

theorem Th38:
  f is_integrable_on M implies r(#)f is_integrable_on M & Integral
  (M,r(#)f) = r * Integral(M,f)
proof
A1: Integral(M,r(#)Re(f)) = Integral(M,Re(r(#)f)) by Th2;
A2: Integral(M,r(#)Im(f)) = Integral(M,Im(r(#)f)) by Th2;
  assume
A3: f is_integrable_on M;
  then
A4: Re f is_integrable_on M;
  then
A5: Integral(M,Re f) < +infty by MESFUNC6:90;
  r(#)Re(f) is_integrable_on M by A4,MESFUNC6:102;
  then
A6: Re(r(#)f) is_integrable_on M by Th2;
  then
A7: Integral(M,Re(r(#)f)) < +infty by MESFUNC6:90;
A8: Im f is_integrable_on M by A3;
  then
A9: -infty < Integral(M,Im f) by MESFUNC6:90;
A10: Integral(M,Im f) < +infty by A8,MESFUNC6:90;
  -infty < Integral(M,Re f) by A4,MESFUNC6:90;
  then reconsider R1=Integral(M,Re f), I1=Integral(M,Im f)
as Element of REAL by A5,A9,A10
,XXREAL_0:14;
A11: (r qua ExtReal) * R1 = r * R1;
A12: (r qua ExtReal) * I1 = r * I1;
  r(#)Im(f) is_integrable_on M by A8,MESFUNC6:102;
  then
A13: Im(r(#)f) is_integrable_on M by Th2;
  then
A14: -infty < Integral(M,Im(r(#)f)) by MESFUNC6:90;
A15: Integral(M,Im(r(#)f)) < +infty by A13,MESFUNC6:90;
  -infty < Integral(M,Re(r(#)f)) by A6,MESFUNC6:90;
  then reconsider
  R2=Integral(M,Re(r(#)f)), I2=Integral(M,Im(r(#)f))
as Element of REAL by A7,A14,A15,
XXREAL_0:14;
A16: Integral(M,r(#)Im(f)) =  r * Integral(M,Im f) by A8,MESFUNC6:102;
A17: r(#)f is_integrable_on M by A6,A13;
  Integral(M,r(#)Re(f))
    = (r qua ExtReal) * Integral(M,Re f) by A4,MESFUNC6:102;
  then R2 + I2 * <i> = r * (R1 + I1 * <i>) by A16,A1,A2,A11,A12;
  then Integral(M,r(#)f) = r * (R1 + I1 * <i>) by A17,Def3;
  hence thesis by A3,A6,A13,Def3;
end;
