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

theorem Th10:
   for S being RingExtension of R, 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, 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;
       p.n = p1.n & q.n = q2.n by A1; then
A3:    [p1.n,q2.n] in [:the carrier of R,the carrier of R:] by ZFMISC_1:def 2;
       thus (p+q).n = p.n + q.n by NORMSP_1:def 2
       .= ((the addF of S)||the carrier of R).(p1.n,q2.n) by A1,A2,C0SP1:def 3
       .= p1.n + q2.n by A3,FUNCT_1:49
       .= (p1+q2).n by NORMSP_1:def 2;
     end;
     hence thesis;
   end;
