let X be set ; :: thesis: for seq being Complex_Sequence
for h being PartFunc of COMPLEX ,COMPLEX st rng seq c= dom (h | X) & h " {0 } = {} holds
((h ^ ) | X) /* seq = ((h | X) /* seq) "
let seq be Complex_Sequence; :: thesis: for h being PartFunc of COMPLEX ,COMPLEX st rng seq c= dom (h | X) & h " {0 } = {} holds
((h ^ ) | X) /* seq = ((h | X) /* seq) "
let h be PartFunc of COMPLEX ,COMPLEX ; :: thesis: ( rng seq c= dom (h | X) & h " {0 } = {} implies ((h ^ ) | X) /* seq = ((h | X) /* seq) " )
assume that
A1: rng seq c= dom (h | X) and
A2: h " {0 } = {} ; :: thesis: ((h ^ ) | X) /* seq = ((h | X) /* seq) "

now

let x be set ; :: thesis: ( x in rng seq implies x in dom ((h ^ ) | X) )
assume x in rng seq ; :: thesis: x in dom ((h ^ ) | X)
then x in dom (h | X) by A1;
then A3: x in (dom h) /\ X by RELAT_1:90;
then x in (dom h) \ (h " {0c }) by A2, XBOOLE_0:def 4;
then A4: x in dom (h ^ ) by CFUNCT_1:def 2;
x in X by A3, XBOOLE_0:def 4;
then x in (dom (h ^ )) /\ X by A4, XBOOLE_0:def 4;
hence x in dom ((h ^ ) | X) by RELAT_1:90; :: thesis: verum

end;

then A5: rng seq c= dom ((h ^ ) | X) by TARSKI:def 3;

now

let n be Element of NAT ; :: thesis: (((h ^ ) | X) /* seq) . n = (((h | X) /* seq) " ) . n
A6: seq . n in rng seq by VALUED_0:28;
then seq . n in dom (h | X) by A1;
then A7: seq . n in (dom h) /\ X by RELAT_1:90;
then seq . n in (dom h) \ (h " {0c }) by A2, XBOOLE_0:def 4;
then A8: seq . n in dom (h ^ ) by CFUNCT_1:def 2;
seq . n in X by A7, XBOOLE_0:def 4;
then seq . n in (dom (h ^ )) /\ X by A8, XBOOLE_0:def 4;
then A9: seq . n in dom ((h ^ ) | X) by RELAT_1:90;
thus (((h ^ ) | X) /* seq) . n = ((h ^ ) | X) /. (seq . n) by A5, FUNCT_2:186
.= (h ^ ) /. (seq . n) by A9, PARTFUN2:32
.= (h /. (seq . n)) " by A8, CFUNCT_1:def 2
.= ((h | X) /. (seq . n)) " by A1, A6, PARTFUN2:32
.= (((h | X) /* seq) . n) " by A1, FUNCT_2:186
.= (((h | X) /* seq) " ) . n by VALUED_1:10 ; :: thesis: verum

end;

hence ((h ^ ) | X) /* seq = ((h | X) /* seq) " by FUNCT_2:113; :: thesis: verum