theorem Th23:
  f in L1_Functions M & g in L1_Functions M implies f + g in L1_Functions M
proof
  set W = L1_Functions M;
  assume that
A1: f in W and
A2: g in W;
  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 by A1;
  then consider NDv be Element of S such that
A3: M.NDv=0 and
A4: dom f = NDv` and
A5: f is_integrable_on M;
  ex g1 be PartFunc of X,REAL st g1=g &ex ND be Element of S st M.ND=0 &
  dom g1 = ND` & g1 is_integrable_on M by A2;
  then consider NDu be Element of S such that
A6: M.NDu=0 and
A7: dom g = NDu` and
A8: g is_integrable_on M;
A9: dom (f+g)= NDv` /\ NDu` by A4,A7,VALUED_1:def 1
    .= (NDv \/ NDu)` by XBOOLE_1:53;
  M.(NDv \/ NDu) =0 & f+g is_integrable_on M by A3,A5,A6,A8,Lm4,MESFUNC6:100;
  hence thesis by A9;
end;
