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 & g is_integrable_on M & B c= dom(f+g) implies f+
g is_integrable_on M & Integral_on(M,B,f+g) = Integral_on(M,B,f) + Integral_on(
  M,B,g)
proof
  assume that
A1: f is_integrable_on M & g is_integrable_on M and
A2: B c= dom(f+g);
A3: dom(f+g) = dom(R_EAL(f+g)) .= dom(R_EAL f + R_EAL g) by Th23;
A4: R_EAL f is_integrable_on M & R_EAL g is_integrable_on M by A1;
  then R_EAL f + R_EAL g is_integrable_on M by A2,A3,MESFUNC5:111;
  then
A5: R_EAL(f+g) is_integrable_on M by Th23;
  Integral_on(M,B,R_EAL f + R_EAL g) = Integral_on(M,B,R_EAL f) +
  Integral_on(M,B,R_EAL g) by A2,A4,A3,MESFUNC5:111;
  hence thesis by A5,Th23;
end;
