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) by VALUED_1:def 4;
then ( x in dom f & x in dom g ) 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 A2, VALUED_1:5; :: thesis: verum