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;
reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,REAL,
  r for Real,
  E,A,B for Element of S;

theorem
  f is_integrable_on M implies f|B is_integrable_on M & Integral_on(M,B,
  r(#)f) = r * Integral_on(M,B,f)
proof
  assume f is_integrable_on M;
  then
A1: R_EAL f is_integrable_on M;
  then Integral_on(M,B,r(#)(R_EAL f)) = r * Integral_on(M,B,R_EAL f) by
MESFUNC5:112;
  then
A2: Integral_on(M,B,R_EAL(r(#)f)) = r * Integral_on(M,B,R_EAL f) by Th20;
  R_EAL(f|B) is_integrable_on M by A1,MESFUNC5:112;
  hence thesis by A2;
end;
