
theorem Th55:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 f be PartFunc of X,ExtREAL, E be Element of S
 st E c= dom f & f is E-measurable holds
  Integral(M,(-f)|E) = - Integral(M,f|E)
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
    f be PartFunc of X,ExtREAL, E be Element of S;
    assume that
A1:  E c= dom f and
A2:  f is E-measurable;
A3: E = dom(f|E) by A1,RELAT_1:62; then
A4: E = dom f /\ E by RELAT_1:61;
    (-f)|E = -(f|E) by Th3;
    hence Integral(M,(-f)|E) = - Integral(M,f|E)
      by A3,A4,A2,MESFUNC5:42,Th52;
end;
