let r be Real; :: according to PARTFUN3:def 1 :: 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 positive Real by Def1;

a " is positive ;

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 by Th2;

then f . x in rng f by A1, FUNCT_1:def 3;

then reconsider a = f . x as positive Real by Def1;

a " is positive ;

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