
theorem Th5:
  for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
      f be PartFunc of X,ExtREAL holds
  max+f is nonnegative & max-f is nonnegative &
    |.f.| is nonnegative
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
    f be PartFunc of X,ExtREAL;
A1: for x be object st x in dom max- f holds 0<= (max-f).x by MESFUNC2:13;
    for x be object st x in dom max+ f holds 0<= (max+f).x by MESFUNC2:12;
    hence max+f is nonnegative & max-f is nonnegative by A1,SUPINF_2:52;
    now
     let x be object;
     assume x in dom |.f.|;
     then (|.f.|).x =|. f.x .| by MESFUNC1:def 10;
     hence 0 <= (|.f.|).x by EXTREAL1:14;
    end;
    hence thesis by SUPINF_2:52;
end;
