
theorem Th83:
  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 (ex E be Element of S
  st E = dom f & f is E-measurable ) & f is nonnegative & A c= B holds
  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;
  assume that
A1: ex E be Element of S st E = dom f & f is E-measurable and
A2: f is nonnegative and
A3: A c= B;
  set A9 = A /\ B;
A4: A9 = A by A3,XBOOLE_1:28;
  set B9 = B \ A;
A5: A9\/B9 = B by XBOOLE_1:51;
  integral+(M,f|(A9\/B9)) =integral+(M,f|A9)+integral+(M,f|B9) by A1,A2,Th81,
XBOOLE_1:89;
  hence thesis by A1,A2,A4,A5,Th80,XXREAL_3:39;
end;
