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

theorem Th49:
  ( ex f be PartFunc of X,REAL st f in L1_Functions M & x=
a.e-eq-class(f,M) & ||.x.|| = Integral(M,abs f) ) & for f be PartFunc of X,REAL
  st f in x holds Integral(M,abs f) = ||.x.||
proof
  reconsider y=x as Point of Pre-L-Space M;
  consider f be PartFunc of X,REAL such that
A1: f in y and
A2: (L-1-Norm M).y = Integral(M,abs f) by Def19;
  y in the carrier of Pre-L-Space M;
  then y in CosetSet M by Def18;
  then consider g be PartFunc of X,REAL such that
A3: y=a.e-eq-class(g,M) & g in L1_Functions M;
  g in y by A3,Th38;
  then f a.e.= g,M by A1,Th46;
  then x = a.e-eq-class(f,M) by A1,A3,Th39;
  hence thesis by A1,A2,Th48;
end;
