reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  F for sequence of S,
  f,g for PartFunc of X,REAL,
  A,B for Element of S,
  r,s for Real,
  a for Real,
  n for Nat;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S,
  r for Real,
  p for Rational;
reserve X for non empty set,
  f,g for PartFunc of X,REAL,
  r for Real ;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S;
reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  f,g,h for PartFunc of X,REAL,
  A for Element of S,
  r for Real;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL ,
  A,B for Element of S,
  r,s for Real;

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;
