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;

theorem Th43:
  f in L1_Functions M & g in L1_Functions M & f a.e.= g,M implies
  Integral(M,f) = Integral(M,g)
proof
  assume that
A1: f in L1_Functions M and
A2: g in L1_Functions M and
A3: f a.e.= 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,REAL 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;
  R_EAL f is_integrable_on M by A6;
  then consider E1 being Element of S such that
A10: E1 = dom R_EAL f and
A11: R_EAL f is E1-measurable;
A12: f is E1-measurable by A11;
A13: ex g1 be PartFunc of X,REAL 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
A14: M.NDg=0 and
A15: dom g = NDg` and
  g is_integrable_on M;
A16: M.(EQ \/ NDg) = 0 by A14,A4,Lm4;
  R_EAL g is_integrable_on M by A13;
  then consider E2 being Element of S such that
A17: E2 = dom R_EAL g and
A18: R_EAL g is E2-measurable;
A19: g is E2-measurable by A18;
A20: EQ` \ (NDf \/NDg) = (EQ \/ (NDf \/NDg))` by XBOOLE_1:41
    .=(NDg \/ (EQ \/NDf))` by XBOOLE_1:4
    .=NDg` \ (EQ \/NDf) by XBOOLE_1:41;
A21: EQ` \ (NDf \/NDg) = (EQ \/ (NDf \/NDg))` by XBOOLE_1:41
    .= (NDf \/ (EQ \/NDg))` by XBOOLE_1:4
    .= NDf` \ (EQ \/NDg) by XBOOLE_1:41;
A22: 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 A22,FUNCT_1:51;
  hence Integral(M,f) = Integral(M,g|(NDg` \(EQ\/NDf))) by A8,A10,A12,A21,A20
,A16,MESFUNC6:89
    .= Integral(M,g) by A15,A17,A19,A9,MESFUNC6:89;
end;
