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) holds
abs ((h | X) /* seq) = ((abs 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) holds
abs ((h | X) /* seq) = ((abs h) | X) /* seq
let h be PartFunc of W,REAL; :: thesis: for seq being sequence of W st rng seq c= dom (h | X) holds
abs ((h | X) /* seq) = ((abs h) | X) /* seq
let seq be sequence of W; :: thesis: ( rng seq c= dom (h | X) implies abs ((h | X) /* seq) = ((abs h) | X) /* seq )
assume A1: rng seq c= dom (h | X) ; :: thesis: abs ((h | X) /* seq) = ((abs h) | X) /* seq
A2: dom (h | X) = (dom h) /\ X by RELAT_1:61
.= (dom (abs h)) /\ X by VALUED_1:def 11
.= dom ((abs h) | X) by RELAT_1:61 ;

now :: thesis: for n being Element of NAT holds (abs ((h | X) /* seq)) . n = (((abs h) | X) /* seq) . n

let n be Element of NAT ; :: thesis: (abs ((h | X) /* seq)) . n = (((abs h) | X) /* seq) . n
A3: seq . n in rng seq by VALUED_0:28;
then seq . n in dom (h | X) by A1;
then seq . n in (dom h) /\ X by RELAT_1:61;
then A4: seq . n in X by XBOOLE_0:def 4;
thus (abs ((h | X) /* seq)) . n = |.(((h | X) /* seq) . n).| by SEQ_1:12
.= |.((h | X) . (seq . n)).| by A1, FUNCT_2:108
.= |.(h . (seq . n)).| by A1, A3, FUNCT_1:47
.= (abs h) . (seq . n) by VALUED_1:18
.= ((abs h) | X) . (seq . n) by A4, FUNCT_1:49
.= (((abs h) | X) /* seq) . n by A1, A2, FUNCT_2:108 ; :: thesis: verum

end;

hence abs ((h | X) /* seq) = ((abs h) | X) /* seq by FUNCT_2:63; :: thesis: verum