theorem Th41:
  f in L1_Functions M & f1 in L1_Functions M & g in L1_Functions M
& g1 in L1_Functions M & a.e-eq-class(f,M) = a.e-eq-class(f1,M) & a.e-eq-class(
  g,M) = a.e-eq-class(g1,M) implies a.e-eq-class(f+g,M) = a.e-eq-class(f1+g1,M)
proof
  assume that
A1: f in L1_Functions M & f1 in L1_Functions M & g in L1_Functions M &
  g1 in L1_Functions M and
A2: a.e-eq-class(f,M) = a.e-eq-class(f1,M) & a.e-eq-class(g,M) =
  a.e-eq-class(g1,M);
  f a.e.= f1,M & g a.e.= g1,M by A1,A2,Th39;
  then
A3: f + g a.e.= f1+g1,M by Th31;
  f + g in L1_Functions M & f1+g1 in L1_Functions M by A1,Th23;
  hence thesis by A3,Th39;
end;
