A1: rng (f ^ g) = (rng f) \/ (rng g) by AFINSQ_1:26;
( not {} in rng f & not {} in rng g ) by RELAT_1:def 9;
then not {} in rng (f ^ g) by A1, XBOOLE_0:def 3;
hence f ^ g is non-empty by RELAT_1:def 9; :: thesis: verum