
theorem Th39:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 f be PartFunc of X,REAL, 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,REAL, A be Element of S;
    assume that
A1:  A = dom f and
A2:  f is A-measurable;
    dom(R_EAL f) = dom f by MESFUNC5:def 7; then
    Integral(M,-(R_EAL f)) = -Integral(M,R_EAL f)
      by A1,A2,MESFUNC6:def 1,MESFUN11:52; then
    Integral(M,R_EAL(-f)) = - Integral(M,R_EAL f) by MESFUNC6:28; then
    Integral(M,-f) = - Integral(M,R_EAL f) by MESFUNC6:def 3;
    hence Integral(M,-f) = - Integral(M,f) by MESFUNC6:def 3;
end;
