let r be Real; :: according to PARTFUN3:def 4 :: thesis: ( r in rng (f ^) implies 0 <= r )

set R = f ^ ;

assume r in rng (f ^) ; :: thesis: 0 <= r

then consider x being object such that

A1: x in dom (f ^) and

A2: (f ^) . x = r by FUNCT_1:def 3;

dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def 2;

then x in dom f by A1, XBOOLE_0:def 5;

then f . x in rng f by FUNCT_1:def 3;

then reconsider a = f . x as non negative Real by Def4;

not a " is negative ;

hence 0 <= r by A1, A2, RFUNCT_1:def 2; :: thesis: verum

set R = f ^ ;

assume r in rng (f ^) ; :: thesis: 0 <= r

then consider x being object such that

A1: x in dom (f ^) and

A2: (f ^) . x = r by FUNCT_1:def 3;

dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def 2;

then x in dom f by A1, XBOOLE_0:def 5;

then f . x in rng f by FUNCT_1:def 3;

then reconsider a = f . x as non negative Real by Def4;

not a " is negative ;

hence 0 <= r by A1, A2, RFUNCT_1:def 2; :: thesis: verum