theorem Th18:
  f in L1_CFunctions M implies a(#)f in L1_CFunctions M
proof
  set W = L1_CFunctions M;
  assume f in W; then
    ex f1 be PartFunc of X,COMPLEX 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,MESFUN6C:40,
    VALUED_1:def 5;
  hence thesis by A1;
end;
