let r be Real; :: according to PARTFUN3:def 1 :: thesis: ( r in rng (sqrt f) implies 0 < r )
set R = sqrt f;
assume r in rng (sqrt f) ; :: thesis: 0 < r
then consider x being object such that
A1: x in dom (sqrt f) and
A2: (sqrt f) . x = r by FUNCT_1:def 3;
dom (sqrt f) = dom f by Def5;
then f . x in rng f by A1, FUNCT_1:def 3;
then reconsider a = f . x as positive Real by Def1;
sqrt a is positive ;
hence 0 < r by A1, A2, Def5; :: thesis: verum