set R = f - g;
let r be real number ; :: according to PARTFUN3:def 1 :: 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_1:12;
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 non positive number by A3, Def3;
a - b is positive ;
hence 0 < r by A1, A2, VALUED_1:13; :: thesis: verum