 reserve K,F,E for Field,
         R,S for Ring;

theorem Th12:
   for S being RingExtension of R
   for p,q being Polynomial of R
   for p1,q1 being Polynomial of S st p = p1 & q = q1 holds p *' q = p1 *' q1
   proof
     let S be RingExtension of R;
     let p,q be Polynomial of R; let p1,q2 be Polynomial of S;
     assume
A1:  p = p1 & q = q2;
A2:  R is Subring of S by Def1;
     now let n be Element of NAT;
       consider r being FinSequence of the carrier of R such that
A3:    len r = n+1 & (p*'q).n = Sum r &
       for k being Element of NAT st k in dom r
       holds r.k = p.(k-'1) * q.(n+1-'k) by POLYNOM3:def 9;
       consider r1 being FinSequence of the carrier of S such that
A4:    len r1 = n+1 & (p1*'q2).n = Sum r1 &
       for k being Element of NAT st k in dom r1
       holds r1.k = p1.(k-'1) * q2.(n+1-'k) by POLYNOM3:def 9;
A5:    dom r1 = Seg(len r) by A3,A4,FINSEQ_1:def 3
       .= dom r by FINSEQ_1:def 3;
       now let m be Nat;
         assume
A6:      m in dom r;
         p.(m-'1) = p1.(m-'1) & q.(n+1-'m) = q2.(n+1-'m) by A1; then
A7:      [p1.(m-'1),q2.(n+1-'m)] in [:the carrier of R,the carrier of R:]
         by ZFMISC_1:def 2;
         thus
         r.m = p.(m-'1) * q.(n+1-'m) by A6,A3
         .= ((the multF of S)||the carrier of R).(p1.(m-'1),q2.(n+1-'m))
            by A1,A2,C0SP1:def 3
         .= p1.(m-'1) * q2.(n+1-'m) by A7,FUNCT_1:49
         .= r1.m by A6,A5,A4;
       end; then
       r = r1 by A5;
       hence (p*'q).n = (p1*'q2).n by A4,A3,A2,Th2;
     end;
     hence thesis;
   end;
