theorem
  f is_integrable_on M & g is_integrable_on M & f is nonnegative & g is
  nonnegative implies f+g is_integrable_on M
proof
  assume that
A1: f is_integrable_on M & g is_integrable_on M and
A2: f is nonnegative & g is nonnegative;
  R_EAL f is_integrable_on M & R_EAL g is_integrable_on M by A1;
  then R_EAL f + R_EAL g is_integrable_on M by A2,MESFUNC5:106;
  then R_EAL(f+g) is_integrable_on M by Th23;
  hence thesis;
end;
