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 negative Real by Def4;

not r * a is negative ;

hence 0 <= z by A1, A2, VALUED_1:def 5; :: thesis: verum

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 negative Real by Def4;

not r * a is negative ;

hence 0 <= z by A1, A2, VALUED_1:def 5; :: thesis: verum