let X be set ; :: thesis: for W being non empty set
for h being PartFunc of W,REAL
for seq being sequence of W st rng seq c= dom (h | X) & h " = {} holds
((h ^) | X) /* seq = ((h | X) /* seq) "

let W be non empty set ; :: thesis: for h being PartFunc of W,REAL
for seq being sequence of W st rng seq c= dom (h | X) & h " = {} holds
((h ^) | X) /* seq = ((h | X) /* seq) "

let h be PartFunc of W,REAL; :: thesis: for seq being sequence of W st rng seq c= dom (h | X) & h " = {} holds
((h ^) | X) /* seq = ((h | X) /* seq) "

let seq be sequence of W; :: thesis: ( rng seq c= dom (h | X) & h " = {} implies ((h ^) | X) /* seq = ((h | X) /* seq) " )
assume that
A1: rng seq c= dom (h | X) and
A2: h " = {} ; :: thesis: ((h ^) | X) /* seq = ((h | X) /* seq) "
now :: thesis: for x being object st x in rng seq holds
x in dom ((h ^) | X)
let x be object ; :: 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_1:61;
then x in (dom h) \ (h " ) by ;
then A4: x in dom (h ^) by RFUNCT_1:def 2;
x in X by ;
then x in (dom (h ^)) /\ X by ;
hence x in dom ((h ^) | X) by RELAT_1:61; :: thesis: verum
end;
then A5: rng seq c= dom ((h ^) | X) by TARSKI:def 3;
now :: thesis: for n being Element of NAT holds (((h ^) | X) /* seq) . n = (((h | X) /* seq) ") . n
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_1:61;
then A8: seq . n in X by XBOOLE_0:def 4;
seq . n in (dom h) \ (h " ) by ;
then A9: seq . n in dom (h ^) by RFUNCT_1:def 2;
thus (((h ^) | X) /* seq) . n = ((h ^) | X) . (seq . n) by
.= (h ^) . (seq . n) by
.= (h . (seq . n)) " by
.= ((h | X) . (seq . n)) " by
.= (((h | X) /* seq) . n) " by
.= (((h | X) /* seq) ") . n by VALUED_1:10 ; :: thesis: verum
end;
hence ((h ^) | X) /* seq = ((h | X) /* seq) " by FUNCT_2:63; :: thesis: verum