
theorem Th61:
  for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
  f be PartFunc of X,ExtREAL st (ex A be Element of S st A = dom f & f
  is A-measurable) & f is nonpositive holds 0 >= 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;
    assume that
A1:  ex A be Element of S st A = dom f & f is A-measurable and
A2:  f is nonpositive;
    consider A be Element of S such that
A3:  A = dom f and
a3:  f is A-measurable by A1;
A4: A = dom(-f) by A3,MESFUNC1:def 7;
    Integral(M,-f) >= 0 by A4,A2,A3,a3,MEASUR11:63,MESFUNC5:90; then
A7: integral+(M,-f) >= 0 by A4,A2,A3,a3,MEASUR11:63,MESFUNC5:88;
    Integral(M,f) = - integral+(M,-f) by A2,A3,a3,Th57;
    hence 0 >= Integral(M,f) by A7;
end;
