let seq be Complex_Sequence; :: thesis: for h being PartFunc of COMPLEX,COMPLEX st rng seq c= dom (h ^) holds
(h ^) /* seq = (h /* seq) "
let h be PartFunc of COMPLEX,COMPLEX; :: thesis: ( rng seq c= dom (h ^) implies (h ^) /* seq = (h /* seq) " )
assume A1: rng seq c= dom (h ^) ; :: thesis: (h ^) /* seq = (h /* seq) "
then A2: ( (dom h) \ (h " {0c}) c= dom h & rng seq c= (dom h) \ (h " {0c}) ) by CFUNCT_1:def 2, XBOOLE_1:36;
now :: thesis: for n being Element of NAT holds ((h ^) /* seq) . n = ((h /* seq) ") . n
let n be Element of NAT ; :: thesis: ((h ^) /* seq) . n = ((h /* seq) ") . n
A3: seq . n in rng seq by VALUED_0:28;
thus ((h ^) /* seq) . n = (h ^) /. (seq . n) by A1, FUNCT_2:109
.= (h /. (seq . n)) " by A1, A3, CFUNCT_1:def 2
.= ((h /* seq) . n) " by A2, FUNCT_2:109, XBOOLE_1:1
.= ((h /* seq) ") . n by VALUED_1:10 ; :: thesis: verum
end;
hence (h ^) /* seq = (h /* seq) " by FUNCT_2:63; :: thesis: verum