let z be Real; :: according to PARTFUN3:def 4 :: 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 non positive Real by Def3;
not r * a is negative ;
hence 0 <= z by A1, A2, VALUED_1:def 5; :: thesis: verum