let z be Real; :: according to PARTFUN3:def 2 :: thesis: ( z in rng (r (#) f) implies 0 > z )
set R = r (#) f;
assume z in rng (r (#) f) ; :: thesis: 0 > z
then consider x being object such that
A1: x in dom (r (#) f) and
A2: (r (#) f) . x = z by FUNCT_1:def 3;
dom (r (#) f) = dom f by VALUED_1:def 5;
then f . x in rng f by A1, FUNCT_1:def 3;
then reconsider a = f . x as positive Real by Def1;
r * a is negative ;
hence 0 > z by A1, A2, VALUED_1:def 5; :: thesis: verum