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) by VALUED_1:def 4;

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 positive Real by Def1;

x in dom g by A1, A3, XBOOLE_0:def 4;

then g . x in rng g by FUNCT_1:def 3;

then reconsider b = g . x as negative Real by Def2;

a * b is negative ;

hence 0 > r by A2, VALUED_1:5; :: thesis: verum

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) by VALUED_1:def 4;

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 positive Real by Def1;

x in dom g by A1, A3, XBOOLE_0:def 4;

then g . x in rng g by FUNCT_1:def 3;

then reconsider b = g . x as negative Real by Def2;

a * b is negative ;

hence 0 > r by A2, VALUED_1:5; :: thesis: verum