theorem
  f is_integrable_on M & g is_integrable_on M implies ex E be Element of
  S st E = dom f /\ dom g & Integral(M,f+g)=Integral(M,f|E)+Integral(M,g|E)
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 consider E be Element of S such that
A1: E = dom(R_EAL f) /\ dom(R_EAL g) & Integral(M,R_EAL f + R_EAL g) =
  Integral( M,(R_EAL f)|E) + Integral(M,(R_EAL g)|E) by MESFUNC5:109;
  take E;
  thus thesis by A1,Th23;
end;
