
theorem
  for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
f be PartFunc of X,ExtREAL, A,B be Element of S st f is_integrable_on M & B = (
  dom f)\A holds f|A is_integrable_on M & Integral(M,f) = Integral(M,f|A)+
  Integral(M,f|B)
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be
  PartFunc of X,ExtREAL, A,B be Element of S such that
A1: f is_integrable_on M and
A2: B = dom f \ A;
  A \/ B = A\/dom f by A2,XBOOLE_1:39;
  then
A3: dom f /\ (A \/ B) = dom f by XBOOLE_1:7,28;
A4: f|(A\/B) = f|(dom f)|(A\/B) by GRFUNC_1:23
    .= f|(dom f /\(A\/B)) by RELAT_1:71
    .= f by A3,GRFUNC_1:23;
  A misses B by A2,XBOOLE_1:79;
  hence thesis by A1,A4,Th97,Th98;
end;
