theorem
  f is_simple_func_in S & f is nonnegative implies Integral(M,f) =
  integral+(M,R_EAL f) & Integral(M,f) = integral'(M,R_EAL f)
proof
  assume that
A1: f is_simple_func_in S and
A2: f is nonnegative;
A3: R_EAL f is_simple_func_in S by A1,Th49;
  then reconsider A=dom(R_EAL f) as Element of S by MESFUNC5:37;
  R_EAL f is A-measurable by A3,MESFUNC2:34;
  then f is A-measurable;
  hence Integral(M,f) = integral+(M,R_EAL f) by A2,Th82;
  hence thesis by A2,A3,MESFUNC5:77;
end;
