reserve a,b,r for Real;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,REAL);
reserve u,v,w for VECTOR of RLSp_PFunctA;
reserve X for non empty set,
  x for Element of X,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E,E1,E2 for Element of S,
  f,g,h,f1,g1 for PartFunc of X ,REAL;
reserve v,u for VECTOR of RLSp_L1Funct M;
reserve v,u for VECTOR of RLSp_AlmostZeroFunct M;

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;
