let A be Ring; :: thesis: for B being comRing
for p, q being Polynomial of A
for x being Element of B st A is Subring of B holds
Ext_eval ((p *' q),x) = (Ext_eval (p,x)) * (Ext_eval (q,x))
let B be comRing; :: thesis: for p, q being Polynomial of A
for x being Element of B st A is Subring of B holds
Ext_eval ((p *' q),x) = (Ext_eval (p,x)) * (Ext_eval (q,x))
let p, q be Polynomial of A; :: thesis: for x being Element of B st A is Subring of B holds
Ext_eval ((p *' q),x) = (Ext_eval (p,x)) * (Ext_eval (q,x))
let x be Element of B; :: thesis: ( A is Subring of B implies Ext_eval ((p *' q),x) = (Ext_eval (p,x)) * (Ext_eval (q,x)) )
assume A0: A is Subring of B ; :: thesis: Ext_eval ((p *' q),x) = (Ext_eval (p,x)) * (Ext_eval (q,x))
defpred S₁[ Nat] means for p being Polynomial of A st len p = $1 holds
Ext_eval ((p *' q),x) = (Ext_eval (p,x)) * (Ext_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 A st len p = n holds
Ext_eval ((p *' q),x) = (Ext_eval (p,x)) * (Ext_eval (q,x)) ; :: thesis: S₁[k]
let p be Polynomial of A; :: thesis: ( len p = k implies Ext_eval ((p *' q),x) = (Ext_eval (p,x)) * (Ext_eval (q,x)) )
assume A3: len p = k ; :: thesis: Ext_eval ((p *' q),x) = (Ext_eval (p,x)) * (Ext_eval (q,x))

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

suppose A4: len p <> 0 ; :: thesis: Ext_eval ((p *' q),x) = (Ext_eval (p,x)) * (Ext_eval (q,x))

set LMp = Leading-Monomial p;
consider r being Polynomial of A such that
A5: len r < len p and
A6: p = r + (Leading-Monomial p) and
for n being Element of NAT st n < (len p) - 1 holds
r . n = p . n by A4, POLYNOM4:16;
thus Ext_eval ((p *' q),x) = Ext_eval (((r *' q) + ((Leading-Monomial p) *' q)),x) by A6, POLYNOM3:32
.= (Ext_eval ((r *' q),x)) + (Ext_eval (((Leading-Monomial p) *' q),x)) by A0, Th19
.= ((Ext_eval (r,x)) * (Ext_eval (q,x))) + (Ext_eval (((Leading-Monomial p) *' q),x)) by A2, A3, A5
.= ((Ext_eval (r,x)) * (Ext_eval (q,x))) + ((Ext_eval ((Leading-Monomial p),x)) * (Ext_eval (q,x))) by A0, Th15
.= ((Ext_eval (r,x)) + (Ext_eval ((Leading-Monomial p),x))) * (Ext_eval (q,x)) by VECTSP_1:def 7
.= (Ext_eval (p,x)) * (Ext_eval (q,x)) by A0, A6, Th19 ; :: thesis: verum

end;

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

then A7: p = 0_. A by POLYNOM4:5;
hence Ext_eval ((p *' q),x) = Ext_eval ((0_. A),x) by POLYNOM4:2
.= (0. B) * (Ext_eval (q,x)) by Th17
.= (Ext_eval (p,x)) * (Ext_eval (q,x)) by A7, Th17 ;
:: thesis: verum

end;

end;

end;

A8: for n being Nat holds S₁[n] from NAT_1:sch 4(A1);
len p = len p ;
hence Ext_eval ((p *' q),x) = (Ext_eval (p,x)) * (Ext_eval (q,x)) by A8; :: thesis: verum