dom (bD (f,h)) = (dom (Shift (f,(- h)))) /\ (dom f) by VALUED_1:12
.= REAL /\ (dom f) by FUNCT_2:def 1
.= REAL /\ REAL by FUNCT_2:def 1
.= REAL ;
hence bD (f,h) is quasi_total by FUNCT_2:def 1; :: thesis: verum