reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  F for sequence of S,
  f,g for PartFunc of X,REAL,
  A,B for Element of S,
  r,s for Real,
  a for Real,
  n for Nat;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S,
  r for Real,
  p for Rational;
reserve X for non empty set,
  f,g for PartFunc of X,REAL,
  r for Real ;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S;
reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  f,g,h for PartFunc of X,REAL,
  A for Element of S,
  r for Real;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL ,
  A,B for Element of S,
  r,s for Real;
reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,REAL,
  r for Real,
  E,A,B for Element of S;

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;
