dom (fD (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 fD (f,h) is quasi_total by FUNCT_2:def 1; :: thesis: verum

.= REAL /\ (dom f) by FUNCT_2:def 1

.= REAL /\ REAL by FUNCT_2:def 1

.= REAL ;

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