let n be Ordinal; :: thesis: 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),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 ; :: thesis: for p being Polynomial of n,L

for a being Element of L

for x being Function of n,L holds eval ((p * a),x) = (eval (p,x)) * a

let p be Polynomial of n,L; :: thesis: for a being Element of L

for x being Function of n,L holds eval ((p * a),x) = (eval (p,x)) * a

let a be Element of L; :: thesis: for x being Function of n,L holds eval ((p * a),x) = (eval (p,x)) * a

let x be Function of n,L; :: thesis: eval ((p * a),x) = (eval (p,x)) * a

thus eval ((p * a),x) = eval ((p *' (a | (n,L))),x) by Th28

.= (eval (p,x)) * a by Lm5 ; :: thesis: verum

for p being Polynomial of n,L

for a being Element of L

for x being Function of n,L holds eval ((p * a),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 ; :: thesis: for p being Polynomial of n,L

for a being Element of L

for x being Function of n,L holds eval ((p * a),x) = (eval (p,x)) * a

let p be Polynomial of n,L; :: thesis: for a being Element of L

for x being Function of n,L holds eval ((p * a),x) = (eval (p,x)) * a

let a be Element of L; :: thesis: for x being Function of n,L holds eval ((p * a),x) = (eval (p,x)) * a

let x be Function of n,L; :: thesis: eval ((p * a),x) = (eval (p,x)) * a

thus eval ((p * a),x) = eval ((p *' (a | (n,L))),x) by Th28

.= (eval (p,x)) * a by Lm5 ; :: thesis: verum