let r be Real; :: according to PARTFUN3:def 1 :: 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_1:12;
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 non positive Real by Def3;
a - b is positive ;
hence 0 < r by A1, A2, VALUED_1:13; :: thesis: verum