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 B-measurable & A = dom f /\ B implies f|B is A-measurable
proof
  assume that
A1: f is B-measurable and
A2: A = dom f /\ B;
A3: A = dom Im(f) /\ B by A2,COMSEQ_3:def 4;
  Im f is B-measurable by A1;
  then (Im f)|B is A-measurable by A3,MESFUNC6:76;
  then
A4: Im(f|B) is A-measurable by Th7;
A5: A = dom Re(f) /\ B by A2,COMSEQ_3:def 3;
  Re f is B-measurable by A1;
  then (Re f)|B is A-measurable by A5,MESFUNC6:76;
  then Re(f|B) is A-measurable by Th7;
  hence thesis by A4;
end;
