
theorem Th51:
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 holds
  integral+(M,f|(A\/B)) <= 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;

    set A1 = A \ B;
A3: integral+(M,f|A1) <= integral+(M,f|A) by A1,A2,XBOOLE_1:36,MESFUNC5:83;
A4: A1\/B = A\/B by XBOOLE_1:39;
    integral+(M,f|(A1\/B)) = integral+(M,f|A1) + integral+(M,f|B)
      by A1,A2,MESFUNC5:81,XBOOLE_1:79;
    hence integral+(M,f|(A\/B)) <= integral+(M,f|A) + integral+(M,f|B)
      by A3,A4,XXREAL_3:35;
end;
