let n be Ordinal; for L being non trivial right_complementable Abelian add-associative right_zeroed well-unital distributive associative commutative left_zeroed doubleLoopStr
for p being Polynomial of n,L
for a being Element of L
for x being Function of n,L holds eval ((p *' (a | (n,L))),x) = (eval (p,x)) * a
let L be non trivial right_complementable Abelian add-associative right_zeroed well-unital distributive associative commutative left_zeroed doubleLoopStr ; for p being Polynomial of n,L
for a being Element of L
for x being Function of n,L holds eval ((p *' (a | (n,L))),x) = (eval (p,x)) * a
let p be Polynomial of n,L; for a being Element of L
for x being Function of n,L holds eval ((p *' (a | (n,L))),x) = (eval (p,x)) * a
let a be Element of L; for x being Function of n,L holds eval ((p *' (a | (n,L))),x) = (eval (p,x)) * a
let x be Function of n,L; eval ((p *' (a | (n,L))),x) = (eval (p,x)) * a
thus eval ((p *' (a | (n,L))),x) =
(eval (p,x)) * (eval ((a | (n,L)),x))
by POLYNOM2:25
.=
(eval (p,x)) * a
by Th25
; verum