let r be Real; :: according to PARTFUN3:def 2 :: 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 by Th2;
then f . x in rng f by A1, FUNCT_1:def 3;
then reconsider a = f . x as negative Real by Def2;
a " is negative ;
hence 0 > r by A1, A2, RFUNCT_1:def 2; :: thesis: verum