let r be Real; :: according to PARTFUN3:def 3 :: 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 1;
then x in dom g by A1, XBOOLE_0:def 4;
then A4: g . x in rng g by FUNCT_1:def 3;
x in dom f by A1, A3, XBOOLE_0:def 4;
then f . x in rng f by FUNCT_1:def 3;
then reconsider a = f . x, b = g . x as non positive Real by A4, Def3;
not a + b is positive ;
hence 0 >= r by A1, A2, VALUED_1:def 1; :: thesis: verum