 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 Th40:
  ex NORM be Function of the carrier of Pre-L-CSpace M,REAL st for
  x be Point of Pre-L-CSpace M ex f be PartFunc of X,COMPLEX
  st f in x & NORM.x = Integral(M,abs f)
proof
  defpred P[set,set] means ex f be PartFunc of X,COMPLEX st f in $1 & $2 =
  Integral(M,abs f);
A1: for x be Point of Pre-L-CSpace M ex y being Element of REAL st P[x,y]
  proof
    let x be Point of Pre-L-CSpace M;
    x in the carrier of Pre-L-CSpace M;
    then x in CCosetSet M by Def19;
    then consider f be PartFunc of X,COMPLEX such that
A2: x=a.e-Ceq-class(f,M) and
A3: f in L1_CFunctions M;
    ex f0 be PartFunc of X,COMPLEX st f=f0 & ex ND be Element of S st M.ND=0
    & dom f0 = ND` & f0 is_integrable_on M by A3;
    then Integral(M,abs f) in REAL by Th37;
    hence thesis by A2,A3,Th31;
  end;
  consider f being Function of Pre-L-CSpace M,REAL such that
A4: for x being Point of Pre-L-CSpace M holds P[x,f.x] from FUNCT_2:sch 3
  (A1 );
  take f;
  thus thesis by A4;
end;
