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;

theorem Th47:
  f in x implies f is_integrable_on M & f in L1_Functions M & abs
  f is_integrable_on M
proof
  x in the carrier of Pre-L-Space M;
  then x in CosetSet M by Def18;
  then consider h be PartFunc of X,REAL such that
A1: x=a.e-eq-class(h,M) and
  h in L1_Functions M;
  assume f in x;
  then ex g be PartFunc of X,REAL st f=g & g in L1_Functions M & h in
  L1_Functions M & h a.e.= g,M by A1;
  then
  ex f0 be PartFunc of X,REAL st f=f0 & ex ND be Element of S st M.ND=0 &
  dom f0 = ND` & f0 is_integrable_on M;
  hence thesis by Th44;
end;
