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

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