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;
