theorem
  ( ex A be Element of S st dom f = A ) implies for c be Real, 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;
  let c be Real, B be Element of S;
  assume f is B-measurable;
  then f is (A/\B)-measurable by A1,Th80;
  then
A2: c(#)f is (A/\B)-measurable by A1,Th21,XBOOLE_1:17;
  dom(c(#)f) = A by A1,VALUED_1:def 5;
  hence thesis by A2,Th80;
end;
