A1: rng (R \/ S) = (rng R) \/ (rng S) by RELAT_1:12;
( rng R c= COMPLEX & rng S c= COMPLEX ) by Def1;
hence rng (R \/ S) c= COMPLEX by A1, XBOOLE_1:8; :: according to VALUED_0:def 1 :: thesis: verum