let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL holds
( max+ f is nonnegative & max- f is nonnegative & |.f.| is nonnegative )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL holds
( max+ f is nonnegative & max- f is nonnegative & |.f.| is nonnegative )
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL holds
( max+ f is nonnegative & max- f is nonnegative & |.f.| is nonnegative )
let f be PartFunc of X,ExtREAL; :: thesis: ( max+ f is nonnegative & max- f is nonnegative & |.f.| is nonnegative )
A1: for x being object st x in dom (max- f) holds
0 <= (max- f) . x by MESFUNC2:13;
for x being 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; :: thesis: |.f.| is nonnegative

now :: thesis: for x being object st x in dom |.f.| holds
0 <= |.f.| . x

let x be object ; :: thesis: ( x in dom |.f.| implies 0 <= |.f.| . x )
assume x in dom |.f.| ; :: thesis: 0 <= |.f.| . x
then |.f.| . x = |.(f . x).| by MESFUNC1:def 10;
hence 0 <= |.f.| . x by EXTREAL1:14; :: thesis: verum

end;

hence |.f.| is nonnegative by SUPINF_2:52; :: thesis: verum