 reserve a,b,r for Complex;
 reserve V for ComplexLinearSpace;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,COMPLEX);
reserve u,v,w for VECTOR of CLSp_PFunctA;
reserve X for non empty set,
  x for Element of X,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E,E1,E2,A,B for Element of S,
  f,g,h,f1,g1 for PartFunc of X,COMPLEX;
reserve v,u for VECTOR of CLSp_L1Funct M;
reserve v,u for VECTOR of CLSp_AlmostZeroFunct M;
reserve x for Point of Pre-L-CSpace M;
reserve x,y for Point of L-1-CSpace M;

theorem Th47:
  f in L1_CFunctions M & g in L1_CFunctions M implies
  Integral(M,|.f+g.|) <= Integral(M,|.f.|) + Integral(M,|.g.|)
proof
  assume that
A1: f in L1_CFunctions M and
A2: g in L1_CFunctions M;
    ex f1 be PartFunc of X,COMPLEX st f=f1
    & ex NDf be Element of S st M.NDf=0 &
  dom f1 = NDf` & f1 is_integrable_on M by A1;
  then consider NDf1 be Element of S such that
A3: M.NDf1=0 and
A4: dom f = NDf1` & f is_integrable_on M;
  R_EAL |.f.| is_integrable_on M by A4,Th37,MESFUNC6:def 4;
  then consider Ef being Element of S such that
A5: Ef = dom R_EAL |.f.| and
A6: R_EAL |.f.| is Ef-measurable;
   ex g1 be PartFunc of X,COMPLEX st g=g1
    & ex NDg be Element of S st M.NDg=0 &
  dom g1 = NDg` & g1 is_integrable_on M by A2;
  then consider NDg1 be Element of S such that
A7: M.NDg1=0 and
A8: dom g = NDg1` & g is_integrable_on M;
  R_EAL |.g.| is_integrable_on M by A8,Th37,MESFUNC6:def 4;
  then consider Eg being Element of S such that
A9: Eg = dom R_EAL |.g.| and
A10: R_EAL |.g.| is Eg-measurable;
    consider E be Element of S such that
A11: E = dom(f+g) &
      Integral(M,(|.f+g.|)|E) <= Integral(M,(|.f.|)|E)
      + Integral(M,(|.g.|)|E) by A4,A8,MESFUN7C:42;
A12: dom(|.f+g.|) = E by A11,VALUED_1:def 11;
      set NF = NDf1` /\ NDg1;
      set NG = NDg1` /\ NDf1;
      NDf1` is Element of S & NDg1` is Element of S by MEASURE1:def 1; then
A13:  NF is Element of S & NG is Element of S &
      Ef is Element of S & Eg is Element of S by MEASURE1:11;
A14: Ef = NDf1` by A4,A5,VALUED_1:def 11;
A15: Eg = NDg1` by A8,A9,VALUED_1:def 11;
      then
A16: E = Ef /\ Eg by A11,A14,A4,A8,VALUED_1:def 1;
A17: Ef \Eg = (X \ NDf1) \ X \/ (X \ NDf1) /\ NDg1 by A14,A15,XBOOLE_1:52
    .= X \(NDf1 \/ X) \/ (X \ NDf1) /\ NDg1 by XBOOLE_1:41
    .= X \ X \/(X \ NDf1)/\ NDg1 by XBOOLE_1:12
    .= {} \/ (X \ NDf1) /\ NDg1 by XBOOLE_1:37
    .= NF;
A18: E = Ef \ (Ef \Eg) by A16,XBOOLE_1:48;
A19: Eg \Ef = (X \ NDg1) \ X \/ (X \ NDg1) /\ NDf1 by A14,A15,XBOOLE_1:52
    .= X \(NDg1 \/ X) \/ (X \ NDg1) /\ NDf1 by XBOOLE_1:41
    .= X \ X \/(X \ NDg1)/\ NDf1 by XBOOLE_1:12
    .= {} \/ (X \ NDg1) /\ NDf1 by XBOOLE_1:37
    .= NG;
A20: E = Eg \ (Eg \Ef) by A16,XBOOLE_1:48;
     NDf1 is measure_zero of M & NDg1 is measure_zero of M
     by A3,A7,MEASURE1:def 7; then
     NF is measure_zero of M & NG is measure_zero of M
     by A13,MEASURE1:36,XBOOLE_1:17; then
A21: M.NF = 0 & M.NG = 0 by MEASURE1:def 7;
A22: Integral(M,(|.f.|)|E) = Integral(M,(|.f.|))
     by A18,A17,A6,MESFUNC6:def 1,A5,A21,MESFUNC6:89;
     Integral(M,(|.g.|)|E) = Integral(M,(|.g.|))
     by A10,MESFUNC6:def 1,A9,A21,A20,A19,MESFUNC6:89;
     hence thesis by A11,A12,RELAT_1:69,A22;
end;
