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 Th40:
  f is_integrable_on M implies c(#)f is_integrable_on M & Integral
  (M,c(#)f) = c * Integral(M,f)
proof
A1: c = Re(c) + Im(c) * <i> by COMPLEX1:13;
A2: dom(<i>(#)f) = dom f by VALUED_1:def 5;
  assume
A3: f is_integrable_on M;
  then
A4: Integral(M,(Re c)(#)f) = Re c * Integral(M,f) by Th38;
A5: <i>(#)f is_integrable_on M by A3,Th39;
  then
A6: (Im c)(#)(<i>(#)f) is_integrable_on M by Th38;
A7: (Re c)(#)f is_integrable_on M by A3,Th38;
  then consider E be Element of S such that
A8: E = dom ((Re c)(#)f) /\ dom ((Im c)(#)(<i>(#)f)) and
A9: Integral(M,(Re c)(#)f + (Im c)(#)(<i>(#)f)) = Integral(M,((Re c)(#)
  f)|E) + Integral(M,((Im c)(#)(<i>(#)f))|E) by A6,Th36;
A10: dom(c(#)f) = dom f by VALUED_1:def 5;
A11: Integral(M,(Im c)(#)(<i>(#)f)) = Im c * Integral(M,<i>(#)f) by A5,Th38;
A12: dom(Re(c)(#)f) = dom f by VALUED_1:def 5;
A13: dom((Im c)(#)(<i>(#)f)) = dom(<i>(#)f) by VALUED_1:def 5;
A14: dom((Re c)(#)f + (Im c)(#)(<i>(#)f)) = dom((Re c)(#)f) /\ dom((Im c)(#)(
  <i>(#)f)) by VALUED_1:def 1;
  now
    let x be Element of X;
    assume
A15: x in dom(c(#)f);
    then
A16: (c(#)f).x = c * f.x by VALUED_1:def 5;
A17: ((Im c)(#)(<i>(#)f)).x = Im c * (<i>(#)f).x by A10,A2,A13,A15,
VALUED_1:def 5;
A18: ((Re c)(#)f).x= Re c * f.x by A10,A12,A15,VALUED_1:def 5;
A19: (<i>(#)f).x = <i> * f.x by A10,A2,A15,VALUED_1:def 5;
    ((Re c)(#)f + (Im c)(#)(<i>(#)f)).x = ((Re c)(#)f).x + ((Im c)(#)(<i>
    (#) f)).x by A10,A12,A2,A13,A14,A15,VALUED_1:def 1;
    hence (c(#)f).x = ((Re c)(#)f + (Im c)(#)(<i>(#)f)).x by A1,A16,A18,A17,A19
;
  end;
  then
A20: c(#)f = (Re c)(#)f + (Im c)(#)(<i>(#)f) by A10,A12,A2,A13,A14,PARTFUN1:5;
  hence c(#)f is_integrable_on M by A7,A6,Th33;
A21: Integral(M,<i>(#)f) = <i> * Integral(M,f) by A3,Th39;
  thus
  Integral(M,c(#)f) = Re c * Integral(M,f) + (Im c)*<i>* Integral(M,f) by A12
,A2,A13,A20,A4,A21,A11,A8,A9
    .= c * Integral(M,f) by A1;
end;
