set R = f / g;
let r be real number ; :: according to PARTFUN3:def 2 :: thesis: ( r in rng (f / g) implies 0 > r )
assume r in rng (f / g) ; :: thesis: 0 > r
then consider x being set such that
A1: x in dom (f / g) and
A2: (f / g) . x = r by FUNCT_1:def 5;
dom (f / g) = (dom f) /\ ((dom g) \ (g " {0 })) by RFUNCT_1:def 4;
then ( x in dom f & x in (dom g) \ (g " {0 }) ) by A1, XBOOLE_0:def 4;
then A3: ( f . x in rng f & g . x in rng g ) by FUNCT_1:def 5;
then reconsider a = f . x as real positive number by Def1;
reconsider b = g . x as real negative number by A3, Def2;
a * (b " ) is negative ;
hence 0 > r by A1, A2, RFUNCT_1:def 4; :: thesis: verum