reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,ExtREAL,
  E for Element of S;
reserve E1,E2 for Element of S;
reserve x,A for set;
reserve a,b for Real;

theorem
  f is_integrable_on M & g is_integrable_on M implies ex E be Element of
S st E = dom(f+g) & Integral(M,(|.f+g.|)|E) <= Integral(M,(|.f.|)|E) + Integral
  (M,(|.g.|)|E)
proof
  assume that
A1: f is_integrable_on M and
A2: g is_integrable_on M;
A3: |.g.| is_integrable_on M by A2,MESFUNC5:100;
A4: f+g is_integrable_on M by A1,A2,MESFUNC5:108;
A5: |.f+g.| is_integrable_on M by A4,MESFUNC5:100;
  for x be Element of X st x in dom |.f+g.| holds (|.f+g.|).x <= (|.f.|+
  |.g.|).x by Th21;
  then
A6: (|.f.|+|.g.|) - |.f+g.| is nonnegative by Th1;
  set G = |.g.|;
  set F = |.f.|;
A7: dom |.f+g.| = dom(f+g) by MESFUNC1:def 10
    .= (dom f /\ dom g) \((f"{-infty} /\ g"{+infty}) \/ (f"{+infty} /\ g"{
  -infty})) by MESFUNC1:def 3;
A8: |.f.| is_integrable_on M by A1,MESFUNC5:100;
  then |.f.|+|.g.| is_integrable_on M by A3,MESFUNC5:108;
  then consider E be Element of S such that
A9: E = dom(|.f.|+|.g.|) /\ dom |.f+g.| and
A10: Integral(M,(|.f+g.|)|E) <= Integral(M,(|.f.|+|.g.|)|E) by A5,A6,Th3;
A11: G|E is_integrable_on M by A3,MESFUNC5:97;
  F|E is_integrable_on M by A8,MESFUNC5:97;
  then consider E1 be Element of S such that
A12: E1 = dom (F|E) /\ dom (G|E) and
A13: Integral(M,F|E+G|E) = Integral(M,(F|E)|E1) + Integral(M,(G|E)|E1)
  by A11,MESFUNC5:109;
  take E;
  dom(G|E) = dom G /\ E by RELAT_1:61;
  then
A14: dom(G|E) = dom g /\ E by MESFUNC1:def 10;
A15: dom (|.f.|+|.g.|) = dom f /\ dom g by Th19;
  then
A16: E = dom |.f+g.| by A9,A7,XBOOLE_1:28,36;
  dom(F|E) = dom F /\ E by RELAT_1:61;
  then dom(F|E) = dom f /\ E by MESFUNC1:def 10;
  then E1 = (dom f /\ E /\ E) /\ dom g by A12,A14,XBOOLE_1:16;
  then E1 = (dom f /\ (E /\ E)) /\ dom g by XBOOLE_1:16;
  then E1 = dom f /\ dom g /\ E by XBOOLE_1:16;
  then
A17: E1 = E by A9,A15,A7,XBOOLE_1:28,36;
  then
A18: (G|E)|E1 = G|E by FUNCT_1:51;
  (F|E)|E1 = F|E by A17,FUNCT_1:51;
  hence thesis by A10,A13,A16,A18,Th20,MESFUNC1:def 10;
end;
