let r be Real; :: according to PARTFUN3:def 1 :: thesis: ( r in rng (abs f) implies 0 < r )
set R = abs f;
assume r in rng (abs f) ; :: thesis: 0 < r
then consider x being object such that
A1: x in dom (abs f) and
A2: (abs f) . x = r by FUNCT_1:def 3;
dom (abs f) = dom f by VALUED_1:def 11;
then reconsider a = f . x as non zero Real by A1, ORDINAL1:def 16;
|.a.| is positive by COMPLEX1:47;
hence 0 < r by A2, VALUED_1:18; :: thesis: verum