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
  f is A-measurable & g is A-measurable & A c= dom g implies f-g
  is A-measurable
proof
  assume that
A1: f is A-measurable and
A2: g is A-measurable and
A3: A c= dom g;
A4: Im g is A-measurable by A2;
A5: A c= dom Re g by A3,COMSEQ_3:def 3;
A6: Re g is A-measurable by A2;
A7: A c= dom Im g by A3,COMSEQ_3:def 4;
  Im f is A-measurable by A1;
  then Im f - Im g is A-measurable by A4,A7,MESFUNC6:29;
  then
A8: Im(f-g) is A-measurable by Th6;
  Re f is A-measurable by A1;
  then Re f - Re g is A-measurable by A6,A5,MESFUNC6:29;
  then Re(f-g) is A-measurable by Th6;
  hence thesis by A8;
end;
