set R = f ^ ;
let r be real number ; :: according to PARTFUN3:def 1 :: thesis: ( r in rng (f ^ ) implies 0 < r )
assume r in rng (f ^ ) ; :: thesis: 0 < r
then consider x being set such that
A1: x in dom (f ^ ) and
A2: (f ^ ) . x = r by FUNCT_1:def 5;
dom (f ^ ) = dom f by Th2;
then f . x in rng f by A1, FUNCT_1:def 5;
then reconsider a = f . x as real positive number by Def1;
a " is positive ;
hence 0 < r by A1, A2, RFUNCT_1:def 8; :: thesis: verum