A1: dom (.: ) = bool A by Def1;
.: is Function of (bool A),(bool (rng R)) by Th21;
then A2: rng (.: ) c= bool (rng R) by RELAT_1:def 19;
bool (rng R) c= bool B by ZFMISC_1:67;
then rng (.: ) c= bool B by A2;
hence .: (R,A) is Function of (bool A),(bool B) by A1, FUNCT_2:def 1, RELSET_1:4; :: thesis: verum