let i be Nat; :: thesis: for V being RealLinearSpace
for v being VECTOR of V
for F being FinSequence of V
for f being Function of the carrier of V,REAL st i in dom F & v = F . i holds
(f (#) F) . i = (f . v) * v

let V be RealLinearSpace; :: thesis: for v being VECTOR of V
for F being FinSequence of V
for f being Function of the carrier of V,REAL st i in dom F & v = F . i holds
(f (#) F) . i = (f . v) * v

let v be VECTOR of V; :: thesis: for F being FinSequence of V
for f being Function of the carrier of V,REAL st i in dom F & v = F . i holds
(f (#) F) . i = (f . v) * v

let F be FinSequence of V; :: thesis: for f being Function of the carrier of V,REAL st i in dom F & v = F . i holds
(f (#) F) . i = (f . v) * v

let f be Function of the carrier of V,REAL; :: thesis: ( i in dom F & v = F . i implies (f (#) F) . i = (f . v) * v )
assume that
A1: i in dom F and
A2: v = F . i ; :: thesis: (f (#) F) . i = (f . v) * v
A3: F /. i = F . i by ;
len (f (#) F) = len F by Def7;
then i in dom (f (#) F) by ;
hence (f (#) F) . i = (f . v) * v by A2, A3, Def7; :: thesis: verum