 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 Th38:
  f in L1_CFunctions M & g in L1_CFunctions M & f a.e.cpfunc= g,M implies
  |.f.| a.e.= |.g.|,M & 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;
  reconsider AF = |. f .|, AG = |. g .| as PartFunc of X,REAL;
A4: 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
A5: M.NDf=0 and
A6: dom f = NDf` and
  f is_integrable_on M;
A7: AF is_integrable_on M by A4,Th37;
  consider EQ being Element of S such that
A8: M.EQ = 0 and
A9: f|EQ` = g|EQ` by A3;
A10:  AF|EQ` = abs(g|EQ`) by A9,CFUNCT_1:54
    .= AG|EQ` by CFUNCT_1:54;
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: AG is_integrable_on M by A11,Th37;
  dom AG = NDg` by A13,VALUED_1:def 11;
  then
A15: AG in L1_Functions M by A12,A14;
  dom AF = NDf` by A6,VALUED_1:def 11;
  then AF in L1_Functions M by A5,A7;
  hence thesis by A15,A8,A10,LPSPACE1:def 10,LPSPACE1:43;
end;
