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

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