 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;

theorem Th41:
  f in x implies f is_integrable_on M & f in L1_CFunctions M & abs
  f is_integrable_on M
proof
  x in the carrier of Pre-L-CSpace M;
  then x in CCosetSet M by Def19;
  then consider h be PartFunc of X,COMPLEX such that
A1: x=a.e-Ceq-class(h,M) and
  h in L1_CFunctions M;
  assume f in x;
  then ex g be PartFunc of X,COMPLEX st f=g & g in L1_CFunctions M & h in
  L1_CFunctions M & h a.e.cpfunc= g,M by A1;
  then
  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;
  hence thesis by Th37;
end;
