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
  ( ex A be Element of S st dom f = A ) implies for c be Complex,
  B be Element of S st f is B-measurable holds c(#)f is B-measurable
proof
  assume ex A be Element of S st A = dom f;
  then consider A be Element of S such that
A1: A = dom f;
  hereby
    let c be Complex, B be Element of S;
A2: dom((Re c)(#)(Re f)) = dom Re f by VALUED_1:def 5;
A3: dom((Im c)(#)(Im f)) = dom Im f by VALUED_1:def 5;
    dom(Re(c(#)f)) = dom((Re c)(#)(Re f) - (Im c)(#)(Im f)) by Th3;
    then
A4: dom(Re(c(#)f)) = dom((Re c)(#)(Re f)) /\ dom((Im c)(#)(Im f)) by
VALUED_1:12;
A5: dom((Im c)(#)(Re f)) = dom Re f by VALUED_1:def 5;
    dom(Im(c(#)f)) = dom((Im c)(#)(Re f) + (Re c)(#)(Im f)) by Th3;
    then
A6: dom(Im(c(#)f)) = dom((Im c)(#)(Re f)) /\ dom((Re c)(#)(Im f)) by
VALUED_1:def 1;
A7: dom((Re c)(#)(Im f)) = dom Im f by VALUED_1:def 5;
A8: dom Re f = dom f by COMSEQ_3:def 3;
A9: dom Im f = dom f by COMSEQ_3:def 4;
    assume
A10: f is B-measurable;
    then Im f is B-measurable;
    then
A11: Im f is (A/\B)-measurable by A1,A9,MESFUNC6:80;
    Re f is B-measurable by A10;
    then Re f is (A/\B)-measurable by A1,A8,MESFUNC6:80;
    then f is (A/\B)-measurable by A11;
    then
A12: c(#)f is (A/\B)-measurable by A1,Th17,XBOOLE_1:17;
    then Im(c(#)f) is (A/\B)-measurable;
    then
A13: Im(c(#)f) is B-measurable by A1,A8,A9,A5,A7,A6,MESFUNC6:80;
    Re(c(#)f) is (A/\B)-measurable by A12;
    then Re(c(#)f) is B-measurable by A1,A8,A9,A2,A3,A4,MESFUNC6:80;
    hence c(#)f is B-measurable by A13;
  end;
end;
