theorem
  f is_integrable_on M & g is_integrable_on M implies f+g is_integrable_on M
proof
  assume f is_integrable_on M & g is_integrable_on M;
  then R_EAL f is_integrable_on M & R_EAL g is_integrable_on M;
  then R_EAL f + R_EAL g is_integrable_on M by MESFUNC5:108;
  then R_EAL(f+g) is_integrable_on M by Th23;
  hence thesis;
end;
