dom (bD (f,h)) =
(dom (Shift (f,(- h)))) /\ (dom f)
by VFUNCT_1:def 2

.= the carrier of V /\ (dom f) by FUNCT_2:def 1

.= the carrier of V /\ the carrier of V by FUNCT_2:def 1

.= the carrier of V ;

hence bD (f,h) is quasi_total by FUNCT_2:def 1; :: thesis: verum

.= the carrier of V /\ (dom f) by FUNCT_2:def 1

.= the carrier of V /\ the carrier of V by FUNCT_2:def 1

.= the carrier of V ;

hence bD (f,h) is quasi_total by FUNCT_2:def 1; :: thesis: verum