let M be non empty set ; :: thesis: for V being ComplexNormSpace
for f being PartFunc of M,V
for z being Complex
for X being set holds (z (#) f) | X = z (#) (f | X)
let V be ComplexNormSpace; :: thesis: for f being PartFunc of M,V
for z being Complex
for X being set holds (z (#) f) | X = z (#) (f | X)
let f be PartFunc of M,V; :: thesis: for z being Complex
for X being set holds (z (#) f) | X = z (#) (f | X)
let z be Complex; :: thesis: for X being set holds (z (#) f) | X = z (#) (f | X)
let X be set ; :: thesis: (z (#) f) | X = z (#) (f | X)

A1: now :: thesis: for c being Element of M st c in dom ((z (#) f) | X) holds
((z (#) f) | X) /. c = (z (#) (f | X)) /. c

let c be Element of M; :: thesis: ( c in dom ((z (#) f) | X) implies ((z (#) f) | X) /. c = (z (#) (f | X)) /. c )
assume A2: c in dom ((z (#) f) | X) ; :: thesis: ((z (#) f) | X) /. c = (z (#) (f | X)) /. c
then A3: c in (dom (z (#) f)) /\ X by RELAT_1:61;
then A4: c in X by XBOOLE_0:def 4;
A5: c in dom (z (#) f) by A3, XBOOLE_0:def 4;
then c in dom f by Def2;
then c in (dom f) /\ X by A4, XBOOLE_0:def 4;
then A6: c in dom (f | X) by RELAT_1:61;
then A7: c in dom (z (#) (f | X)) by Def2;
thus ((z (#) f) | X) /. c = (z (#) f) /. c by A2, PARTFUN2:15
.= z * (f /. c) by A5, Def2
.= z * ((f | X) /. c) by A6, PARTFUN2:15
.= (z (#) (f | X)) /. c by A7, Def2 ; :: thesis: verum

end;

dom ((z (#) f) | X) = (dom (z (#) f)) /\ X by RELAT_1:61
.= (dom f) /\ X by Def2
.= dom (f | X) by RELAT_1:61
.= dom (z (#) (f | X)) by Def2 ;
hence (z (#) f) | X = z (#) (f | X) by A1, PARTFUN2:1; :: thesis: verum