
theorem Th52:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 f be PartFunc of X,ExtREAL, A be Element of S
   st A = dom f & f is A-measurable holds Integral(M,-f) = - 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, A be Element of S;
    assume that
A1:  A = dom f and
A2:  f is A-measurable;
    set g = -f;
A4: f = -g by Th36;
B6: dom(max-f) = A & dom(max+f) = A by A1,MESFUNC2:def 2,def 3;
B7: max-f is A-measurable & max+f is A-measurable by A1,A2,Th10;
A6: dom g = A by A1,MESFUNC1:def 7; then
A7: dom(max+g) = A & dom(max-g) = A by MESFUNC2:def 2,def 3;
    g is A-measurable by A1,A2,MEASUR11:63; then
A9: max+g is A-measurable & max-g is A-measurable by A6,Th10; then
P1: integral+(M,max+g) = Integral(M,max+g) by A7,Th5,MESFUNC5:88
     .= Integral(M,max-(-g)) by Th34
     .= integral+(M,max-f) by A4,B6,B7,Th5,MESFUNC5:88;
    integral+(M,max-g) = Integral(M,max-g) by A7,A9,Th5,MESFUNC5:88
     .= Integral(M,max+(-g)) by MESFUNC2:14
     .= integral+(M,max+f) by A4,B6,B7,Th5,MESFUNC5:88; then
    Integral(M,f)
      = integral+(M,max-g) - integral+(M,max+g) by P1,MESFUNC5:def 16
     .= -(integral+(M,max+g) - integral+(M,max-g)) by XXREAL_3:26;
    hence thesis by MESFUNC5:def 16;
end;
