 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;

theorem Th36:
  f in L1_CFunctions M & g in L1_CFunctions M & f a.e.cpfunc= g,M implies
  Integral(M,f) = Integral(M,g)
proof
  assume that
A1: f in L1_CFunctions M and
A2: g in L1_CFunctions M and
A3: f a.e.cpfunc= g,M;
  consider EQ being Element of S such that
A4: M.EQ = 0 and
A5: f|EQ` = g|EQ` by A3;
A6: ex f1 be PartFunc of X,COMPLEX st f=f1 & ex ND be Element of S st M.ND=0 &
  dom f1 = ND` & f1 is_integrable_on M by A1;
  then consider NDf be Element of S such that
A7: M.NDf=0 and
A8: dom f = NDf` and
  f is_integrable_on M;
A9: M.(EQ \/ NDf) = 0 by A7,A4,Lm4;
     consider E1 being Element of S such that
A10: E1 = dom f and f is E1-measurable by A6,MESFUN7C:35;
A11: ex g1 be PartFunc of X,COMPLEX st g=g1 & ex ND be Element of S st M.ND=0
     & dom g1 = ND` & g1 is_integrable_on M by A2;
  then consider NDg be Element of S such that
A12: M.NDg=0 and
A13: dom g = NDg` and
  g is_integrable_on M;
A14: M.(EQ \/ NDg) = 0 by A12,A4,Lm4;
   consider E2 being Element of S such that
A15: E2 = dom g and g is E2-measurable by A11,MESFUN7C:35;
A16: EQ` \ (NDf \/NDg) = (EQ \/ (NDf \/NDg))` by XBOOLE_1:41
    .=(NDg \/ (EQ \/NDf))` by XBOOLE_1:4
    .=NDg` \ (EQ \/NDf) by XBOOLE_1:41;
A17: EQ` \ (NDf \/NDg) = (EQ \/ (NDf \/NDg))` by XBOOLE_1:41
    .= (NDf \/ (EQ \/NDg))` by XBOOLE_1:4
    .= NDf` \ (EQ \/NDg) by XBOOLE_1:41;
A18: EQ` \ (NDf \/NDg) c= EQ` by XBOOLE_1:36;
  then f|(EQ` \ (NDf \/NDg)) = g|EQ`|(EQ`\(NDf \/NDg)) by A5,FUNCT_1:51
    .= g|(EQ`\(NDf \/ NDg)) by A18,FUNCT_1:51;
  hence Integral(M,f) = Integral(M,g|(NDg` \(EQ\/NDf)))
  by A6,A8,A10,A17,A16,A14,MESFUN6C:22
    .= Integral(M,g) by A11,A13,A15,A9,MESFUN6C:22;
end;
