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;
