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;
