
theorem
  for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
  f be PartFunc of X,ExtREAL st f is_simple_func_in S & f is nonnegative holds
  Integral(M,f) = integral+(M,f) & Integral(M,f) = integral'(M,f)
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be
  PartFunc of X,ExtREAL;
  assume that
A1: f is_simple_func_in S and
A2: f is nonnegative;
  reconsider A=dom f as Element of S by A1,Th37;
  f is A-measurable by A1,MESFUNC2:34;
  hence Integral(M,f) =integral+(M,f) by A2,Th88;
  hence thesis by A1,A2,Th77;
end;
