let r be Real; :: 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 object 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 non positive Real by Def3;

not a " is positive ;

hence 0 >= r by A2, VALUED_1:10; :: thesis: verum

set R = Inv f;

assume r in rng (Inv f) ; :: thesis: 0 >= r

then consider x being object 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 non positive Real by Def3;

not a " is positive ;

hence 0 >= r by A2, VALUED_1:10; :: thesis: verum