let L be non empty non trivial right_complementable Abelian add-associative right_zeroed associative commutative distributive left_unital doubleLoopStr ; :: thesis: for p, q being Polynomial of L
for x being Element of L holds eval (p *' q),x = (eval p,x) * (eval q,x)
let p, q be Polynomial of L; :: thesis: for x being Element of L holds eval (p *' q),x = (eval p,x) * (eval q,x)
let x be Element of L; :: thesis: eval (p *' q),x = (eval p,x) * (eval q,x)
defpred S₁[ Nat] means for p being Polynomial of L st len p = $1 holds
eval (p *' q),x = (eval p,x) * (eval q,x);
A1: for k being Nat st ( for n being Nat st n < k holds
S₁[n] ) holds
S₁[k]

proof

let k be Nat; :: thesis: ( ( for n being Nat st n < k holds
S₁[n] ) implies S₁[k] )
assume A2: for n being Nat st n < k holds
for p being Polynomial of L st len p = n holds
eval (p *' q),x = (eval p,x) * (eval q,x) ; :: thesis: S₁[k]
let p be Polynomial of L; :: thesis: ( len p = k implies eval (p *' q),x = (eval p,x) * (eval q,x) )
assume A3: len p = k ; :: thesis: eval (p *' q),x = (eval p,x) * (eval q,x)

per cases ( len p <> 0 or len p = 0 ) ;

suppose len p <> 0 ; :: thesis: eval (p *' q),x = (eval p,x) * (eval q,x)

then consider r being Polynomial of L such that
A4: len r < len p and
A5: p = r + (Leading-Monomial p) and
for n being Element of NAT st n < (len p) - 1 holds
r . n = p . n by Th19;
set LMp = Leading-Monomial p;
thus eval (p *' q),x = eval ((r *' q) + ((Leading-Monomial p) *' q)),x by A5, POLYNOM3:33
.= (eval (r *' q),x) + (eval ((Leading-Monomial p) *' q),x) by Th22
.= ((eval r,x) * (eval q,x)) + (eval ((Leading-Monomial p) *' q),x) by A2, A3, A4
.= ((eval r,x) * (eval q,x)) + ((eval (Leading-Monomial p),x) * (eval q,x)) by Th26
.= ((eval r,x) + (eval (Leading-Monomial p),x)) * (eval q,x) by VECTSP_1:def 18
.= (eval p,x) * (eval q,x) by A5, Th22 ; :: thesis: verum

end;

suppose len p = 0 ; :: thesis: eval (p *' q),x = (eval p,x) * (eval q,x)

then A6: p = 0_. L by Th8;
hence eval (p *' q),x = eval (0_. L),x by Th5
.= 0. L by Th20
.= (0. L) * (eval q,x) by VECTSP_1:39
.= (eval p,x) * (eval q,x) by A6, Th20 ;
:: thesis: verum

end;

end;

end;

A7: len p = len p ;
for n being Nat holds S₁[n] from NAT_1:sch 4(A1);
hence eval (p *' q),x = (eval p,x) * (eval q,x) by A7; :: thesis: verum