let X be non empty set ; :: thesis: for f being PartFunc of X,COMPLEX holds |.f.| is nonnegative
let f be PartFunc of X,COMPLEX; :: 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 VALUED_1:def 11;
hence 0 <= |.f.| . x by COMPLEX1:46; :: thesis: verum

end;

hence |.f.| is nonnegative by MESFUNC6:52; :: thesis: verum