 reserve a,b,r for Complex;
 reserve V for ComplexLinearSpace;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,COMPLEX);
reserve u,v,w for VECTOR of CLSp_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,A,B for Element of S,
  f,g,h,f1,g1 for PartFunc of X,COMPLEX;
reserve v,u for VECTOR of CLSp_L1Funct M;
reserve v,u for VECTOR of CLSp_AlmostZeroFunct M;

theorem Th34:
  f in L1_CFunctions M & f1 in L1_CFunctions M & g in L1_CFunctions M
& g1 in L1_CFunctions M
& a.e-Ceq-class(f,M) = a.e-Ceq-class(f1,M) & a.e-Ceq-class(g,M)
= a.e-Ceq-class(g1,M) implies a.e-Ceq-class(f+g,M) = a.e-Ceq-class(f1+g1,M)
proof
  assume that
A1: f in L1_CFunctions M & f1 in L1_CFunctions M & g in L1_CFunctions M &
  g1 in L1_CFunctions M and
A2: a.e-Ceq-class(f,M) = a.e-Ceq-class(f1,M) & a.e-Ceq-class(g,M) =
  a.e-Ceq-class(g1,M);
  f a.e.cpfunc= f1,M & g a.e.cpfunc= g1,M by A1,A2,Th32;
  then
A3: f + g a.e.cpfunc= f1+g1,M by Th25;
  f + g in L1_CFunctions M & f1+g1 in L1_CFunctions M by A1,Th17;
  hence thesis by A3,Th32;
end;
