let r be real number ; :: according to PARTFUN3:def 3 :: thesis: ( r in rng (Inv f) implies 0 >= r )
set R = Inv f;
assume r in rng (Inv f) ; :: thesis: 0 >= r
then consider x being set such that
A1: x in dom (Inv f) and
A2: (Inv f) . x = r by FUNCT_1:def 3;
dom (Inv f) = X by FUNCT_2:def 1
.= dom f by FUNCT_2:def 1 ;
then f . x in rng f by A1, FUNCT_1:def 3;
then reconsider a = f . x as real non positive number by Def3;
not a " is positive ;
hence 0 >= r by A2, VALUED_1:10; :: thesis: verum