
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,E be Element of S
  st E = dom f & f is E-measurable & f is nonpositive & 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,E be Element of S;
    assume that
A1:  E = dom f and
A2:  f is E-measurable and
A3:  f is nonpositive and
A4:  A c= B;
    set g = -f;
    E = dom g by A1,MESFUNC1:def 7; then
A5: Integral(M,g|A) <= Integral(M,g|B) by A1,A2,A3,A4,MESFUNC5:93,MEASUR11:63;
    reconsider E1 = E /\ A as Element of S;
A6: dom(f|A) = E1 by A1,RELAT_1:61;
A7: E1 = dom f /\ E1 by A1,XBOOLE_1:17,28;
A8: f is E1-measurable by A2,XBOOLE_1:17,MESFUNC1:30;
A9: f|E1 = f|E|A by RELAT_1:71;
    g|A = -(f|A) by Th3; then
A10:Integral(M,g|A) = -Integral(M,f|A) by A1,A6,A7,A8,A9,Th52,MESFUNC5:42;
    reconsider E2 = E /\ B as Element of S;
A11:dom(f|B) = E2 by A1,RELAT_1:61;
A12:E2 = dom f /\ E2 by A1,XBOOLE_1:17,28;
A13:f is E2-measurable by A2,XBOOLE_1:17,MESFUNC1:30;
A14:f|E2 = f|E|B by RELAT_1:71;
    g|B = -(f|B) by Th3; then
    Integral(M,g|B) = -Integral(M,f|B)
      by A1,A11,A12,A13,A14,Th52,MESFUNC5:42;
    hence Integral(M,f|A) >= Integral(M,f|B) by A5,A10,XXREAL_3:38;
end;
