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

theorem
   for R being Subring of S, p being Polynomial of R holds
   p is Polynomial of S
   proof
     let R be Subring of S, p be Polynomial of R;
A2:  the carrier of R c= the carrier of S by C0SP1:def 3;
A3:  0.S = 0.R by C0SP1:def 3;
     rng p c= the carrier of R by RELAT_1:def 19; then
     rng p c= the carrier of S by A2; then
     reconsider p1 = p as sequence of (the carrier of S) by FUNCT_2:6;
A4:  Support p is finite by POLYNOM1:def 5;
     now let o be object;
       assume
A5:    o in Support p1; then
       reconsider n = o as Element of NAT;
A6:    0.R <> p.n by A3,A5,POLYNOM1:def 3;
       dom p = NAT by FUNCT_2:def 1;
       hence o in Support p by A6,POLYNOM1:def 3;
     end; then
     Support p1 c= Support p;
     hence thesis by A4,POLYNOM1:def 5;
   end;
