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

theorem Th11:
   for S being RingExtension of R holds
   the addF of Polynom-Ring R
   = (the addF of Polynom-Ring S)||the carrier of Polynom-Ring R
   proof
     let S be RingExtension of R;
     set aR = the addF of Polynom-Ring R,
     aS = (the addF of Polynom-Ring S)||the carrier of Polynom-Ring R;
     set cR = the carrier of Polynom-Ring R,
     cS = the carrier of Polynom-Ring S;
A1:  cR c= cS by Th6;
A2:  dom aS = dom(the addF of Polynom-Ring S) /\ [:cR,cR:] by RELAT_1:61
     .= [:cS,cS:] /\ [:cR,cR:] by FUNCT_2:def 1
     .= [:cR,cR:] by A1,ZFMISC_1:96,XBOOLE_1:28
     .= dom aR by FUNCT_2:def 1;
     now let o be object;
       assume
A3:    o in dom aR; then
       consider p,q being object such that
A4:    p in the carrier of Polynom-Ring R &
       q in the carrier of Polynom-Ring R & o = [p,q] by ZFMISC_1:def 2;
       reconsider p,q as Element of cR by A4;
       reconsider p1 = p, q1 = q as Element of cS by A1;
       reconsider p2 = p, q2 = q as Polynomial of R;
       reconsider p3 = p1, q3 = q1 as Polynomial of S;
       thus
       aR.o = p + q by A4
       .= p2 + q2 by POLYNOM3:def 10
       .= p3 + q3 by Th10
       .= p1 + q1 by POLYNOM3:def 10
       .= aS.o by A2,A3,A4,FUNCT_1:47;
     end;
     hence thesis by A2;
   end;
