let r be Real; :: according to PARTFUN3:def 4 :: thesis: ( r in rng (f / f) implies 0 <= r )
set R = f / f;
assume r in rng (f / f) ; :: thesis: 0 <= r
then consider x being object such that
A1: ( x in dom (f / f) & (f / f) . x = r ) by FUNCT_1:def 3;
not (f . x) * ((f . x) ") is negative ;
hence 0 <= r by A1, RFUNCT_1:def 1; :: thesis: verum