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

theorem Th13:
   S is RingExtension of R implies  the multF of Polynom-Ring R
   = (the multF of Polynom-Ring S)||the carrier of Polynom-Ring R
   proof
     assume
AS:  S is RingExtension of R;
     set mR = the multF of Polynom-Ring R,
     mS = (the multF 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 AS,Th6;
A2:  dom mS = dom(the multF 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 mR by FUNCT_2:def 1;
     now let o be object;
       assume
A3:    o in dom mR; then
       consider p,q being object such that
A4:    p in cR & q in cR & 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
       mR.o = p * q by A4
       .= p2 *' q2 by POLYNOM3:def 10
       .= p3 *' q3 by AS,Th12
       .= p1 * q1 by POLYNOM3:def 10
       .= mS.o by A4,A2,A3,FUNCT_1:47;
     end;
     hence thesis by A2;
   end;
