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,
  k for Real,
  n for Nat,
  E for Element of S;

theorem Th31:
  for f,g be PartFunc of X,REAL st dom f /\ dom g = E & f
  is E-measurable & g is E-measurable holds f(#)g is E-measurable
proof
  let f,g be PartFunc of X,REAL;
  assume that
A1: dom f /\ dom g = E and
A2: f is E-measurable & g is E-measurable;
  R_EAL f is E-measurable & R_EAL g is E-measurable by A2,MESFUNC6:def
1;
  then (R_EAL f)(#)(R_EAL g) is E-measurable by A1,MESFUNC7:15;
  then R_EAL(f(#)g) is E-measurable by Th30;
  hence thesis by MESFUNC6:def 1;
end;
