let L be Z_Lattice; :: thesis: for f being Function of (),INT.Ring
for F being FinSequence of ()
for v, u being Vector of ()
for i being Nat st i in dom F & u = F . i holds
(ScFS (v,f,F)) . i = () . (v,((f . u) * u))

let f be Function of (),INT.Ring; :: thesis: for F being FinSequence of ()
for v, u being Vector of ()
for i being Nat st i in dom F & u = F . i holds
(ScFS (v,f,F)) . i = () . (v,((f . u) * u))

let F be FinSequence of (); :: thesis: for v, u being Vector of ()
for i being Nat st i in dom F & u = F . i holds
(ScFS (v,f,F)) . i = () . (v,((f . u) * u))

let v, u be Vector of (); :: thesis: for i being Nat st i in dom F & u = F . i holds
(ScFS (v,f,F)) . i = () . (v,((f . u) * u))

let i be Nat; :: thesis: ( i in dom F & u = F . i implies (ScFS (v,f,F)) . i = () . (v,((f . u) * u)) )
assume A1: ( i in dom F & u = F . i ) ; :: thesis: (ScFS (v,f,F)) . i = () . (v,((f . u) * u))
A2: F /. i = F . i by ;
len (ScFS (v,f,F)) = len F by defScFSDM;
then i in dom (ScFS (v,f,F)) by ;
hence (ScFS (v,f,F)) . i = () . (v,((f . u) * u)) by ; :: thesis: verum