
theorem Th84:
  for X be non empty set, S be SigmaField of X, M be sigma_Measure
of S, f be PartFunc of X,ExtREAL, E,A be Element of S st f is nonnegative & E =
dom f & f is E-measurable & M.A =0 holds integral+(M,f|(E\A)) = 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, E,A be Element of S such that
A1: f is nonnegative and
A2: E = dom f and
A3: f is E-measurable and
A4: M.A =0;
  set B = E\A;
  A \/ B = A \/ E by XBOOLE_1:39;
  then
A5: dom f = dom f /\ (A\/B) by A2,XBOOLE_1:7,28
    .= dom(f|(A\/B)) by RELAT_1:61;
  for x be object st x in dom(f|(A\/B)) holds (f|(A\/B)).x = f.x
          by FUNCT_1:47;
  then
A6: f|(A\/B) =f by A5,FUNCT_1:2;
  integral+(M,f|(A\/B)) =integral+(M,f|A)+integral+(M,f|B) by A1,A2,A3,Th81,
XBOOLE_1:79;
  then integral+(M,f) = 0.+ integral+(M,f|B) by A1,A2,A3,A4,A6,Th82;
  hence thesis by XXREAL_3:4;
end;
