
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,E be Element of S
  st E = dom f & f is E-measurable & f is nonpositive
  holds 0 >= Integral(M,f|A)
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,E be Element of S;
    assume that
A1:  E = dom f and
A2:  f is E-measurable and
A3:  f is nonpositive;
    reconsider E1 = E /\ A as Element of S;
A4: dom(f|A) = E1 by A1,RELAT_1:61;
A5: E1 = dom f /\ E1 by A1,XBOOLE_1:17,28;
    f is E1-measurable by A2,XBOOLE_1:17,MESFUNC1:30; then
A6: f|E1 is E1-measurable by A5,MESFUNC5:42;
    f|E1 = f|E|A by RELAT_1:71;
    hence 0>= Integral(M,f|A) by A1,A3,A4,A6,Th61,Th1;
end;
