 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;
reserve x for Point of Pre-L-CSpace M;
reserve x,y for Point of L-1-CSpace M;

theorem Th43:
  ( ex f be PartFunc of X,COMPLEX st f in L1_CFunctions M & x=
a.e-Ceq-class(f,M) & ||.x.|| = Integral(M,abs f) ) &
for f be PartFunc of X,COMPLEX
  st f in x holds Integral(M,abs f) = ||.x.||
proof
  reconsider y=x as Point of Pre-L-CSpace M;
  consider f be PartFunc of X,COMPLEX such that
A1: f in y and
A2: (L-1-CNorm M).y = Integral(M,abs f) by Def20;
  y in the carrier of Pre-L-CSpace M;
  then y in CCosetSet M by Def19;
  then consider g be PartFunc of X,COMPLEX such that
A3: y=a.e-Ceq-class(g,M) & g in L1_CFunctions M;
  g in y by A3,Th31;
  then f a.e.cpfunc= g,M by A1,Th39;
  then x = a.e-Ceq-class(f,M) by A1,A3,Th32;
  hence thesis by A1,A2,Th42;
end;
