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

let x be object ; :: thesis: ( x in dom (abs f) implies 0 <= (abs f) . x )
assume x in dom (abs f) ; :: thesis: 0 <= (abs f) . x
then (abs f) . x = |.(f . x).| by VALUED_1:def 11;
hence 0 <= (abs f) . x by COMPLEX1:46; :: thesis: verum

end;

hence abs f is nonnegative by MESFUNC6:52; :: thesis: verum