let r be Real; :: according to PARTFUN3:def 2 :: thesis: ( r in rng (f / g) implies 0 > r )
set R = f / g;
assume r in rng (f / g) ; :: thesis: 0 > r
then consider x being object such that
A1: x in dom (f / g) and
A2: (f / g) . x = r by FUNCT_1:def 3;
A3: dom (f / g) = (dom f) /\ ((dom g) \ (g " {0})) by RFUNCT_1:def 1;
then x in dom f by A1, XBOOLE_0:def 4;
then f . x in rng f by FUNCT_1:def 3;
then reconsider a = f . x as negative Real by Def2;
x in (dom g) \ (g " {0}) by A1, A3, XBOOLE_0:def 4;
then g . x in rng g by FUNCT_1:def 3;
then reconsider b = g . x as positive Real by Def1;
a * (b ") is negative ;
hence 0 > r by A1, A2, RFUNCT_1:def 1; :: thesis: verum