theorem Th24:
  f in L1_Functions M implies a(#)f in L1_Functions M
proof
  set W = L1_Functions M;
  assume f in W;
  then
  ex f1 be PartFunc of X,REAL st f1=f & ex ND be Element of S st M.ND=0 &
  dom f1 = ND` & f1 is_integrable_on M;
  then consider ND be Element of S such that
A1: M.ND=0 and
A2: dom f = ND` & f is_integrable_on M;
  dom (a(#)f) = ND` & a(#)f is_integrable_on M by A2,MESFUNC6:102
,VALUED_1:def 5;
  hence thesis by A1;
end;
