theorem
  f is_integrable_on M & g is_integrable_on M & B c= dom(f+g) implies f+
g is_integrable_on M & Integral_on(M,B,f+g) = Integral_on(M,B,f) + Integral_on(
  M,B,g)
proof
  assume that
A1: f is_integrable_on M & g is_integrable_on M and
A2: B c= dom(f+g);
A3: dom(f+g) = dom(R_EAL(f+g)) .= dom(R_EAL f + R_EAL g) by Th23;
A4: 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,A3,MESFUNC5:111;
  then
A5: R_EAL(f+g) is_integrable_on M by Th23;
  Integral_on(M,B,R_EAL f + R_EAL g) = Integral_on(M,B,R_EAL f) +
  Integral_on(M,B,R_EAL g) by A2,A4,A3,MESFUNC5:111;
  hence thesis by A5,Th23;
end;
