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 Th17:
  f is A-measurable & A c= dom f implies c(#)f is A-measurable
proof
  assume that
A1: f is A-measurable and
A2: A c= dom f;
A3: dom Im f = dom f by COMSEQ_3:def 4;
A4: Im f is A-measurable by A1;
  then
A5: Re(c)(#)Im(f) is A-measurable by A2,A3,MESFUNC6:21;
A6: Im(c)(#)Im(f) is A-measurable by A2,A4,A3,MESFUNC6:21;
A7: dom Re f = dom f by COMSEQ_3:def 3;
A8: Re f is A-measurable by A1;
  then Im(c)(#)Re(f) is A-measurable by A2,A7,MESFUNC6:21;
  then Im(c)(#)Re(f) + Re(c)(#)Im(f) is A-measurable by A5,MESFUNC6:26;
  then
A9: Im(c(#)f) is A-measurable by Th3;
  dom(Im(c)(#)Im(f)) = dom Im(f) by VALUED_1:def 5;
  then
A10: A c= dom(Im(c)(#)Im(f)) by A2,COMSEQ_3:def 4;
  Re(c)(#)Re(f) is A-measurable by A2,A8,A7,MESFUNC6:21;
  then Re(c)(#)Re(f) - Im(c)(#)Im(f) is A-measurable by A6,A10,MESFUNC6:29;
  then Re(c(#)f) is A-measurable by Th3;
  hence thesis by A9;
end;
