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 Th39:
  f is_integrable_on M implies <i>(#)f is_integrable_on M &
  Integral(M,<i>(#)f) = <i> * Integral(M,f)
proof
A1: Re(<i>(#)f) = -Im(f) by Th4;
  assume
A2: f is_integrable_on M;
  then
A3: Im f is_integrable_on M;
A4: Im(<i>(#)f)= Re f by Th4;
  then
A5: Im(<i>(#)f) is_integrable_on M by A2;
  Re(<i>(#)f) is_integrable_on M by A3,A1,MESFUNC6:102;
  hence <i>(#)f is_integrable_on M by A5;
  then consider R1,I1 be Real such that
A6: R1=Integral(M,Re(<i>(#)f)) and
A7: I1 =Integral(M,Im(<i>(#)f)) and
A8: Integral(M,<i>(#)f)=R1+ I1 * <i> by Def3;
  consider R,I be Real such that
A9: R=Integral(M,Re f) and
A10: I =Integral(M,Im f) and
A11: Integral(M,f)=R+ I * <i> by A2,Def3;
  R1 = (-1) * (I qua ExtReal) by A3,A1,A10,A6,MESFUNC6:102
    .=(-1)*I;
  hence thesis by A4,A9,A11,A7,A8;
end;
