
theorem
  for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
  f,g be PartFunc of X,ExtREAL, B be Element of S st f is_integrable_on M & g
is_integrable_on M & B c= dom(f+g) holds f+g is_integrable_on M & Integral_on(M
  ,B,f+g) = Integral_on(M,B,f) + Integral_on(M,B,g)
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f,g
  be PartFunc of X,ExtREAL, B be Element of S such that
A1: f is_integrable_on M and
A2: g is_integrable_on M and
A3: B c= dom(f+g);
A4: dom(f|B) = dom f /\ B by RELAT_1:61;
  dom(f+g) = (dom f /\ dom g) \ (f"{-infty}/\g"{+infty} \/ f"{+infty}/\g"{
  -infty}) by MESFUNC1:def 3;
  then
A5: dom(f+g) c= dom f /\ dom g by XBOOLE_1:36;
  dom f /\ dom g c= dom f by XBOOLE_1:17;
  then dom(f+g) c= dom f by A5;
  then
A6: dom(f|B) = B by A3,A4,XBOOLE_1:1,28;
A7: Integral_on(M,B,f+g) =Integral(M,f|B+g|B) by A3,Th29;
A8: g|B is_integrable_on M by A2,Th97;
A9: dom(g|B) = dom g /\ B by RELAT_1:61;
  dom f /\ dom g c= dom g by XBOOLE_1:17;
  then dom(f+g) c= dom g by A5;
  then
A10: dom(g|B) = B by A3,A9,XBOOLE_1:1,28;
  f|B is_integrable_on M by A1,Th97;
  hence thesis by A1,A2,A6,A8,A10,A7,Lm13,Th108;
end;
