reserve a,b,r for Real;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,REAL);
reserve u,v,w for VECTOR of RLSp_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 for Element of S,
  f,g,h,f1,g1 for PartFunc of X ,REAL;
reserve v,u for VECTOR of RLSp_L1Funct M;
reserve v,u for VECTOR of RLSp_AlmostZeroFunct M;
reserve x for Point of Pre-L-Space M;
reserve x,y for Point of L-1-Space M;

theorem Th55:
  f is_integrable_on M & g is_integrable_on M implies Integral(M,
  abs(f+g)) <= Integral(M,abs f) + Integral(M,abs g)
proof
  set f1=R_EAL f;
  set g1=R_EAL g;
  assume that
A1: f is_integrable_on M and
A2: g is_integrable_on M;
A3: f1 is_integrable_on M by A1;
  then consider B be Element of S such that
A4: B = dom f1 and
 f1 is B-measurable;
A5: B= dom |.f1.| by A4,MESFUNC1:def 10;
  |.f1.| is_integrable_on M by A3,MESFUNC5:100;
  then
A6: ex E be Element of S st E = dom |.f1.| & |.f1.| is E-measurable;
A7: g1 is_integrable_on M by A2;
  then consider C be Element of S such that
A8: C = dom g1 and
 g1 is C-measurable;
A9: C= dom |.g1.| by A8,MESFUNC1:def 10;
  consider E be Element of S such that
A10: E = dom(f1+g1) and
A11: Integral(M,(|.f1+g1.|)|E) <= Integral(M,(|.f1.|)|E) + Integral(M,(|.
  g1.|)|E) by A3,A7,MESFUNC7:22;
  dom |.f1+g1.| = E by A10,MESFUNC1:def 10;
  then f1+g1 =R_EAL(f+g) & (|.f1+g1.|)|E = |.f1+g1.| by MESFUNC6:23,RELAT_1:68;
  then
A12: Integral(M,(|.f1+g1.|)|E) = Integral(M,|.f+g.|) by MESFUNC6:1;
  |.g1.| is_integrable_on M by A7,MESFUNC5:100;
  then
A13: ex E be Element of S st E = dom |.g1.| & |.g1.| is E-measurable;
A14: E= (dom f1 /\ dom g1)\((f1"{-infty} /\ g1"{+infty}) \/ (f1"{+infty} /\
  g1"{-infty})) by A10,MESFUNC1:def 3;
  then E c= dom g1 by XBOOLE_1:18,36;
  then E c= dom |.g1.| by MESFUNC1:def 10;
  then Integral(M,(|.g1.|)|E) <= Integral(M,(|.g1.|)|C) by A9,A13,MESFUNC5:93;
  then Integral(M,(|.g1.|)|E) <= Integral(M,|.g1.|) by A9,RELAT_1:68;
  then
A15: Integral(M,(|.g1.|)|E) <= Integral(M,|.g.|) by MESFUNC6:1;
  E c= dom f1 by A14,XBOOLE_1:18,36;
  then E c= dom |.f1.| by MESFUNC1:def 10;
  then Integral(M,(|.f1.|)|E) <= Integral(M,(|.f1.|)|B) by A5,A6,MESFUNC5:93;
  then Integral(M,(|.f1.|)|E) <= Integral(M,|.f1.|) by A5,RELAT_1:68;
  then Integral(M,(|.f1.|)|E) <= Integral(M,|.f.|) by MESFUNC6:1;
  then Integral(M,(|.f1.|)|E) + Integral(M,(|.g1.|)|E) <= Integral(M,|.f.|) +
  Integral(M,|.g.|) by A15,XXREAL_3:36;
  hence thesis by A11,A12,XXREAL_0:2;
end;
