let Y be set ; :: thesis: for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V
for p being Real st f | Y is V8() holds
(p (#) f) | Y is V8()
let C be non empty set ; :: thesis: for V being RealNormSpace
for f being PartFunc of C,V
for p being Real st f | Y is V8() holds
(p (#) f) | Y is V8()
let V be RealNormSpace; :: thesis: for f being PartFunc of C,V
for p being Real st f | Y is V8() holds
(p (#) f) | Y is V8()
let f be PartFunc of C,V; :: thesis: for p being Real st f | Y is V8() holds
(p (#) f) | Y is V8()
let p be Real; :: thesis: ( f | Y is V8() implies (p (#) f) | Y is V8() )
assume f | Y is V8() ; :: thesis: (p (#) f) | Y is V8()
then consider r being VECTOR of V such that
A1: for c being Element of C st c in Y /\ (dom f) holds
f /. c = r by PARTFUN2:35;

now

let c be Element of C; :: thesis: ( c in Y /\ (dom (p (#) f)) implies (p (#) f) /. c = p * r )
assume A2: c in Y /\ (dom (p (#) f)) ; :: thesis: (p (#) f) /. c = p * r
then A3: c in Y by XBOOLE_0:def 4;
A4: c in dom (p (#) f) by A2, XBOOLE_0:def 4;
then c in dom f by Def4;
then A5: c in Y /\ (dom f) by A3, XBOOLE_0:def 4;
thus (p (#) f) /. c = p * (f /. c) by A4, Def4
.= p * r by A1, A5 ; :: thesis: verum

end;

hence (p (#) f) | Y is V8() by PARTFUN2:35; :: thesis: verum