A1: rng (R \/ S) = (rng R) \/ (rng S) by RELAT_1:12;
( rng R c= REAL & rng S c= REAL ) by Def3;
hence rng (R \/ S) c= REAL by A1, XBOOLE_1:8; :: according to VALUED_0:def 3 :: thesis: verum