let X be set ; :: thesis: for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V
for r being Real holds (r (#) f) | X = r (#) (f | X)
let C be non empty set ; :: thesis: for V being RealNormSpace
for f being PartFunc of C,V
for r being Real holds (r (#) f) | X = r (#) (f | X)
let V be RealNormSpace; :: thesis: for f being PartFunc of C,V
for r being Real holds (r (#) f) | X = r (#) (f | X)
let f be PartFunc of C,V; :: thesis: for r being Real holds (r (#) f) | X = r (#) (f | X)
let r be Real; :: thesis: (r (#) f) | X = r (#) (f | X)

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

let c be Element of C; :: thesis: ( c in dom ((r (#) f) | X) implies ((r (#) f) | X) /. c = (r (#) (f | X)) /. c )
assume A2: c in dom ((r (#) f) | X) ; :: thesis: ((r (#) f) | X) /. c = (r (#) (f | X)) /. c
then A3: c in (dom (r (#) f)) /\ X by RELAT_1:61;
then A4: c in X by XBOOLE_0:def 4;
A5: c in dom (r (#) f) by A3, XBOOLE_0:def 4;
then c in dom f by Def4;
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 (r (#) (f | X)) by Def4;
thus ((r (#) f) | X) /. c = (r (#) f) /. c by A2, PARTFUN2:15
.= r * (f /. c) by A5, Def4
.= r * ((f | X) /. c) by A6, PARTFUN2:15
.= (r (#) (f | X)) /. c by A7, Def4 ; :: thesis: verum

end;

dom ((r (#) f) | X) = (dom (r (#) f)) /\ X by RELAT_1:61
.= (dom f) /\ X by Def4
.= dom (f | X) by RELAT_1:61
.= dom (r (#) (f | X)) by Def4 ;
hence (r (#) f) | X = r (#) (f | X) by A1, PARTFUN2:1; :: thesis: verum